Grade 3

Estimating sums and differences

The canned food drive took in 287 cans on Monday and 438 on Tuesday. About how many is that, and how can a quick estimate tell you whether a worked-out total is the right size?

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RoundingGrades 3 to 4 Multi-digit additionGrades 2 to 4

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Estimation and mental math5

Estimating sums and differences

Mental strategies2 Estimating by rounding Friendly numbers and deliberate over and under estimates Estimating a huge quantity from small facts

When you reach for it

Iara, Caue and Lucero run their school’s canned food drive. On Monday 287 cans come in, and on Tuesday 438. The principal asks, “About how many so far?” Nobody needs the exact count to answer. “About 700” or “about 730” says it, and either can be worked out in your head in a few seconds.

That is estimating a sum: round each number first, then add the easy rounded numbers. The same move estimates a difference: how many more books the book drive still needs, or how many more cans one class brought than another. The result is an estimate, a number close to the exact answer, written with the symbol \approx, read “is about equal to”:

287 + 438 \approx 730

You reach for an estimate in two situations. Sometimes “about” is all the question needs. And sometimes you have an exact answer, worked out on paper or read off a tally sheet, and you want to know whether it is the right size. An estimate cannot prove an answer right, but it can show quickly that an answer is far too big or far too small.

This lesson uses whole numbers up to 1{,}000 and rounds to the nearest ten or the nearest hundred, as taught in Rounding to the nearest ten or hundred: find the two tens (or hundreds) a number sits between, and take the nearer one, with an exact halfway number rounding up. The exact sums and differences come from Multi-digit addition and Multi-digit subtraction.

The lesson runs over three days. Part 1 estimates sums. Part 2 estimates differences. Part 3 uses estimates to judge whether an exact answer is reasonable.

The procedure

Part 1: estimating a sum

Round each number to the same place. For 287 + 438, rounded to the nearest hundred, 287 becomes 300 (its tens digit 8 puts it past 250) and 438 becomes 400 (its tens digit 3 puts it before 450). Both numbers are rounded to hundreds, so the sum of the rounded numbers is easy: 3 hundreds and 4 hundreds make 7 hundreds.

300 + 400 = 700

Round to tens for a closer estimate. To the nearest ten, 287 becomes 290 and 438 becomes 440. Adding them is still mental work: 2 hundreds and 4 hundreds make 6 hundreds, and 9 tens and 4 tens make 13 tens, which is 1 hundred and 3 tens. So

290 + 440 = 730

Compare with the exact sum. Worked out in full, 287 + 438 = 725. The hundreds estimate is 25 too small. The tens estimate is 5 too big.

Figure 1The exact sum 725 with its two estimates. Rounding to hundreds gives 700, which is 25 away. Rounding to tens gives 730, only 5 away.

Why the estimate stays close. Look at how far rounding moved each number. To the nearest hundred, 287 moved up 13 to reach 300, and 438 moved down 38 to reach 400. Up 13 and down 38 partly cancel, and together they leave the sum 25 lower: 725 - 25 = 700. To the nearest ten, 287 moved up 3 and 438 moved up 2, so the sum moved up 5: 725 + 5 = 730.

Rounding to the nearest ten never moves a number more than 5, because the nearer ten is never more than halfway away. Two numbers each move at most 5, so their sum moves at most 5 + 5 = 10. That gives a promise you can count on:

> Rounding each of two numbers to the nearest ten gives an estimate of their sum that is never > more than 10 away from the exact sum. Rounding to the nearest hundred gives an estimate never > more than 100 away, since each number moves at most 50.

This is why rounding to tens usually gives the closer estimate. The hundreds estimate is quicker, since it only adds single digits of hundreds, but it is allowed to be off by ten times as much. Which one to use depends on what the estimate is for: “about 700 cans” is fine for the principal, and a closer check needs tens.

With three numbers, each one can move up to 5, so a tens estimate of a sum of three numbers is never more than 15 away. Example 2 adds three.

Part 2: estimating a difference

A difference is estimated the same way: round both numbers to the same place, then subtract the rounded numbers.

The book drive has a goal of 812 books and has collected 396. How many more are needed? To the nearest hundred, 812 becomes 800 and 396 becomes 400, and 8 hundreds take away 4 hundreds leaves 4 hundreds:

800 - 400 = 400

To the nearest ten, 812 becomes 810 and 396 becomes 400:

810 - 400 = 410

The exact difference is 812 - 396 = 416, which you can check by adding back: 396 + 416 = 812. The hundreds estimate is 16 too small and the tens estimate is 6 too small.

Figure 2The exact difference 416 with its two estimates. Rounding to hundreds gives 400, which is 16 away; rounding to tens gives 410, which is 6 away.

Why the two moves can add up instead of cancel. In a difference, the two numbers play different parts. To the nearest hundred, the starting number 812 moved down 12, which makes the answer 12 smaller. The amount taken away, 396, moved up 4, and taking away more also makes the answer smaller, by 4. Both moves push the same way, and 12 + 4 = 16. With the tens, 812 moved down 2 and 396 moved up 4, so the answer came out 2 + 4 = 6 too small.

Either way, each number still moves at most 5 when rounded to tens, so the estimate of a difference is never more than 5 + 5 = 10 from the exact difference. Rounded to hundreds, it is never more than 100 away. The same promise as for sums holds.

When the hundreds estimate says almost nothing. If two numbers are close together, they may round to the same hundred, and the hundreds estimate of their difference is 0. That only says the difference is somewhere from 0 up to 100. Rounding to tens is the way to see the gap.

Part 3: is the answer reasonable?

An exact answer is reasonable when it is about the size an estimate says it should be. The promise from Parts 1 and 2 turns this into a test with a clear edge.

Suppose a tally sheet says the sum 563 + 249 is 712. To the nearest ten, 563 becomes 560 and 249 becomes 250, so

563 + 249 \approx 560 + 250 = 810

The exact sum can be no more than 10 from 810, so it lies somewhere from 800 to 820. The number 712 is 98 away from 810, far outside that band. So 712 cannot be the sum, and the addition needs to be done again. Done carefully, 563 + 249 = 812.

Figure 3The tens estimate for 563 + 249 is 810, so the exact sum must lie in the band from 800 to 820. The number 712 is nowhere near it.

The test works in one direction only:

Too far from the estimate means wrong. If a sum or difference of two numbers is more than 10 from the tens estimate, or more than 100 from the hundreds estimate, it cannot be correct.
Close to the estimate means reasonable, not proven. An answer inside the band has passed the size test, but it could still be off by a few. Only the exact work, and a check such as adding back after subtracting, confirms it.

The tens band is 20 wide and the hundreds band is 200 wide. A wrong answer can hide inside the wide band and still be caught by the narrow one, which is the clearest reason to prefer the tens estimate when checking. Example 7 shows exactly that.

Worked examples

Examples 1 and 2 belong to Part 1, Examples 3 and 4 to Part 2, and Examples 5, 6 and 7 to Part 3.

Example 1

Cans on Monday and Tuesday

The canned food drive took in 287 cans on Monday and 438 cans on Tuesday. Estimate the total by rounding to the nearest hundred, then by rounding to the nearest ten. Which estimate is closer to the exact total?

Solution

To the nearest hundred: 287 lies between 200 and 300 and is past the halfway mark 250, so it rounds to 300. 438 lies between 400 and 500 and is before 450, so it rounds to 400. Adding the rounded numbers, 300 + 400 = 700.

To the nearest ten: 287 lies between 280 and 290 with ones digit 7, so it rounds to 290. 438 lies between 430 and 440 with ones digit 8, so it rounds to 440. Adding, 290 + 440 = 730.

The exact total is 287 + 438 = 725 cans. The hundreds estimate is 725 - 700 = 25 away, and the tens estimate is 730 - 725 = 5 away, so the tens estimate is closer. That fits the promise: a tens estimate is never more than 10 from the exact sum.

Example 2

Three classrooms

Iara’s classroom brought 154 cans, Caue’s brought 236 and Lucero’s brought 172. Estimate the total by rounding to the nearest hundred and by rounding to the nearest ten.

Solution

Round every number to the same place, then add all three rounded numbers.

To the nearest hundred: 154 has tens digit 5, so it is past halfway and rounds up to 200. 236 has tens digit 3, so it rounds down to 200. 172 has tens digit 7, so it rounds up to 200. The estimate is 200 + 200 + 200 = 600.

To the nearest ten: 154 rounds to 150, 236 to 240, and 172 to 170. Add two at a time, from left to right: 150 + 240 = 390, and 390 + 170 = 560.

The exact total is 154 + 236 + 172 = 562. The hundreds estimate is 38 too big: rounding pushed 154 up 46 and 172 up 28, and only 236 came down, by 36. The tens estimate is just 2 too small. With three numbers each moving at most 5, a tens estimate is never more than 15 away, and this one is well inside that.

Example 3

How many more books

The book drive’s goal is 812 books, and 396 have been collected. Estimate how many more books are needed, first rounding to the nearest hundred, then to the nearest ten.

Solution

“How many more” is the difference 812 - 396.

To the nearest hundred, 812 rounds down to 800 (tens digit 1), and 396 rounds up to 400 (tens digit 9). The estimate is 800 - 400 = 400.

To the nearest ten, 812 rounds to 810 and 396 rounds up to 400, since it is past the halfway mark 395. The estimate is 810 - 400 = 410.

The exact difference is 812 - 396 = 416; adding back, 396 + 416 = 812, so it is right. The tens estimate, 6 short, is closer than the hundreds estimate, 16 short. Both came out too small for the same reason: the starting number was rounded down and the amount taken away was rounded up, and each of those makes the answer smaller.

Example 4

Two grades, close to halfway

In the book drive, Lucero’s grade collected 451 books and Caue’s grade collected 349. Estimate how many more books Lucero’s grade collected, rounding to the nearest hundred and to the nearest ten. Which estimate is useful here?

Solution

To the nearest hundred, 451 rounds up to 500, since its tens digit is 5, and 349 rounds down to 300, since its tens digit is 4. The estimate is 500 - 300 = 200.

To the nearest ten, 451 rounds to 450 and 349 rounds to 350. The estimate is 450 - 350 = 100.

The exact difference is 451 - 349 = 102. The tens estimate is 2 away. The hundreds estimate is 98 away, almost double the true answer, though still inside its promise of 100.

What went wrong with the hundreds is that both numbers sit very near a halfway mark, so each moved almost 50, and in opposite directions: 451 moved up 49 and 349 moved down 49. Moving the starting number up and the amount taken away down both make the difference bigger, so the two moves added up to 98. Only the tens estimate is useful here.

Example 5

Checking a total on the tally sheet

The tally sheet says that 467 cans and 375 cans make 742 cans in all. Is 742 a reasonable total?

Solution

Estimate the sum by rounding to the nearest ten. 467 rounds to 470, and 375 rounds up to 380, since it sits exactly halfway between 370 and 380. So

467 + 375 \approx 470 + 380 = 850

A tens estimate of a sum of two numbers is never more than 10 from the exact sum, so the exact total is somewhere from 840 to 860. The number 742 is 850 - 742 = 108 below the estimate, far outside that band. 742 is not a reasonable total.

This time the hundreds estimate catches it too. 467 rounds to 500 and 375 rounds to 400, so the sum is about 500 + 400 = 900. The number 742 is 900 - 742 = 158 from that, more than the 100 the hundreds promise allows. An answer this far off is caught by either estimate.

Working the sum again: 467 + 375 = 842, which sits 8 from the tens estimate, as it must. The tally was exactly 100 short: the hundred regrouped from the tens column, 6 tens and 7 tens plus 1 more, was never added in.

Example 6

Checking a difference

At the end of the canned food drive the count is 634 cans, and 278 of them are soup. The tally sheet lists 356 cans that are not soup. Is 356 reasonable?

Solution

The cans that are not soup are the difference 634 - 278. Round to the nearest ten: 634 becomes 630 and 278 becomes 280, so

634 - 278 \approx 630 - 280 = 350

The exact difference is no more than 10 from 350, so it lies from 340 to 360. The listed 356 is 6 above the estimate, inside that band. It is reasonable.

Reasonable is not the same as proven, so finish with an exact check. Adding back, 356 + 278 = 634, which is the full count. The sheet is right.

Rounding to hundreds also lets 356 through: 600 - 300 = 300, and 356 is 56 away, inside the band of 100. But that band runs from 200 to 400, so it would have let many wrong answers through as well.

Example 7

A total the hundreds cannot catch

In the book drive, Iara’s class brought 247 books and Caue’s class brought 365. The tally sheet says the two classes brought 602 books. Check 602 first with a hundreds estimate, then with a tens estimate.

Solution

To the nearest hundred, 247 rounds to 200 (tens digit 4) and 365 rounds to 400 (tens digit 6), so

247 + 365 \approx 200 + 400 = 600

602 is only 2 from 600. It looks very reasonable, and the hundreds test cannot reject it: the exact sum may be anywhere from 500 to 700.

To the nearest ten, 247 rounds to 250 and 365 rounds to 370 (both have ones digit at least 5), so

247 + 365 \approx 250 + 370 = 620

Now the exact sum must lie from 610 to 630. 602 is 620 - 602 = 18 from the estimate, more than 10. So 602 cannot be the sum.

Working it out exactly, 247 + 365 = 612. The closeness of 602 to 600 was luck: rounding to hundreds moved 247 down 47 and 365 up 35, which happened to leave the estimate below the true sum and right beside the wrong one. The tens estimate caught what the hundreds missed.

Boundaries

An estimate is not the answer. 287 + 438 \approx 730, but 287 + 438 = 725. Use the estimate where “about” is enough, and the exact answer where it is not.
Round both numbers to the same place. The promise, within 10 for tens and within 100 for hundreds, comes from each number moving at most 5 or at most 50. Mixing places breaks the promise and also loses the easy adding.
The promise counts the numbers. It is within 10 for two numbers rounded to tens. Each extra number can add up to 5 more, so three numbers give within 15.
Tens are usually closer, not always. A tens estimate is guaranteed to be within 10, and a hundreds estimate only within 100, so the tens estimate is the safer one. But by luck a hundreds estimate can be exact: 245 + 355 = 600, and 200 + 400 = 600, while the tens estimate 250 + 360 = 610 is 10 off.
The reasonableness test only rejects. An answer far from the estimate is wrong. An answer near it is reasonable, and still needs the exact work, or a check such as adding back, to be confirmed.
Whole numbers up to 1{,}000, rounded to tens or hundreds. Larger numbers, other places, and estimating products and quotients come later.

Where this goes next

Estimating by rounding uses the same move on larger numbers and on products, and the window argument from Part 1 still explains how far off an estimate can be. Friendly numbers and deliberate over and under estimates shows when it pays to round toward a friendly number on purpose rather than to the nearest one, and how to tell whether an estimate is over or under. Both sit with this lesson under Estimation and mental math.

The habit itself, asking “is this answer about the right size?” before trusting it, goes with every calculation from here on, from measuring lengths and areas to checking a long subtraction.

Exercises

Part 1: estimating sums

Estimate 342 + 215 by rounding each number to the nearest hundred.
Estimate 176 + 408 by rounding each number to the nearest ten.
Estimate 527 + 289 by rounding to the nearest hundred and then to the nearest ten. The exact sum is 816. Which estimate is closer?
Iara’s class collected 265 cans and Caue’s class collected 193. Estimate the total by rounding to the nearest ten.
Estimate 126 + 341 + 207 by rounding each number to the nearest ten.
Lucero’s group brought 95 cans of beans and 487 cans of soup. Estimate the total by rounding to the nearest hundred and to the nearest ten.

Part 2: estimating differences

Estimate 784 - 312 by rounding each number to the nearest hundred.
Estimate 653 - 247 by rounding each number to the nearest ten.
Estimate 906 - 538 by rounding to the nearest hundred and then to the nearest ten. The exact difference is 368. Which estimate is closer?
The book drive’s goal is 700 books, and 462 have come in. Estimate how many more are needed by rounding to the nearest ten.
Estimate 382 - 317 by rounding to the nearest hundred and to the nearest ten. Find the exact difference, and say how far off each estimate is.
Estimate 436 - 364 by rounding to the nearest hundred and to the nearest ten. What does the hundreds estimate tell you, and what does the tens estimate add?

Part 3: is the answer reasonable?

A tally sheet shows 715 - 286 = 529. Use an estimate to the nearest ten to decide whether 529 is reasonable.
A tally sheet shows 357 + 418 = 775. Use an estimate to the nearest ten to decide whether 775 is reasonable.
A tally sheet shows 603 - 198 = 495. Is 495 reasonable? Use an estimate to the nearest ten.
A tally sheet shows 433 + 457 = 870. Check 870 with an estimate to the nearest hundred and with one to the nearest ten. What does each check say?
The book drive hopes to reach 1{,}000 books. Two grades have brought 468 and 487. Without finding the exact total, use a tens estimate to decide whether the drive has reached its goal. Then find the exact total.

Explain your thinking

Explain why rounding two numbers to the nearest ten always gives an estimate of their sum that is no more than 10 away from the exact sum.
Give two numbers whose hundreds estimate of their sum is exactly 100 more than their exact sum, and explain why 100 is as far off as it can be.
A tally sheet shows 374 + 258 = 628. Show that 628 passes the tens-estimate test, then show that it is still wrong. What does this say about what the test can and cannot do?
Is the tens estimate of a sum always closer than the hundreds estimate? Try 145 + 255.

Answers

500. 342 rounds to 300 and 215 rounds to 200, and 300 + 200 = 500. (Exact: 557.)
590. 176 rounds to 180 and 408 rounds to 410, and 180 + 410 = 590. (Exact: 584.)
Hundreds: 500 + 300 = 800, which is 16 from 816. Tens: 530 + 290 = 820, which is 4 from 816. The tens estimate is closer.
About 460 cans. 265 rounds up to 270 (halfway rounds up) and 193 rounds to 190, and 270 + 190 = 460. (Exact: 458.)
680. The numbers round to 130, 340 and 210; 130 + 340 = 470 and 470 + 210 = 680. (Exact: 674.)
Hundreds: 95 rounds to 100 and 487 to 500, so about 600. Tens: 95 rounds up to 100 and 487 to 490, so about 590. (Exact: 582.)
500. 784 rounds to 800 and 312 to 300, and 800 - 300 = 500. (Exact: 472.)
400. 653 rounds to 650 and 247 to 250, and 650 - 250 = 400. (Exact: 406.)
Hundreds: 900 - 500 = 400, which is 32 from 368. Tens: 910 - 540 = 370, which is 2 from 368. The tens estimate is closer.
About 240 books. 700 stays 700 and 462 rounds to 460, and 700 - 460 = 240. (Exact: 238.)
Hundreds: 400 - 300 = 100. Tens: 380 - 320 = 60. Exact: 382 - 317 = 65. The hundreds estimate is 35 off and the tens estimate is 5 off.
Hundreds: both numbers round to 400, so the estimate is 400 - 400 = 0. That only says the difference is somewhere from 0 to 100. Tens: 440 - 360 = 80, which puts the difference between 70 and 90. (Exact: 72.)
Not reasonable. 720 - 290 = 430, so the exact difference lies from 420 to 440, and 529 is 99 away from 430. (Exact: 715 - 286 = 429.)
Reasonable. 360 + 420 = 780, and 775 is 5 from it, inside the band from 770 to 790. Exact work confirms it: 357 + 418 = 775.
Not reasonable. 600 - 200 = 400, so the exact difference lies from 390 to 410, and 495 is 95 away. (Exact: 603 - 198 = 405.)
Hundreds: 400 + 500 = 900, and 870 is 30 away, inside the band of 100, so this check cannot reject it. Tens: 430 + 460 = 890, and 870 is 20 away, more than 10, so 870 cannot be the sum. (Exact: 433 + 457 = 890.)
Not yet. 470 + 490 = 960, so the exact total is at most 970, short of 1{,}000. Exact: 468 + 487 = 955 books.
A number rounded to the nearest ten moves to the nearer of its two tens, and the nearer ten is never more than halfway, 5, away. So each number moves at most 5, up or down. The sum moves by the two moves together: at most 5 + 5 = 10 if both go the same way, and less if they go opposite ways and partly cancel.
For example, 150 + 250: the hundreds estimate is 200 + 300 = 500 and the exact sum is 400, so the estimate is 100 too big. Each number can move at most 50 when rounded to the nearest hundred, so two numbers move the sum at most 50 + 50 = 100. These two each sit exactly on a halfway mark and both round up the full 50.
Tens: 370 + 260 = 630, and 628 is 2 away, so it passes. But 374 + 258 = 632, so 628 is wrong. The test can show an answer is far too big or too small; it cannot catch an answer that is off by only a little.
No. For 145 + 255 = 400, the hundreds estimate is 100 + 300 = 400, exactly right, while the tens estimate is 150 + 260 = 410, which is 10 off. What rounding to tens does guarantee is being within 10; rounding to hundreds only guarantees within 100.