- For each number, decide which two hundreds it lies between. - Compare its distance from those hundreds before choosing one. - You can add the rounded hundreds without adding all the ones and tens.

1. \(297\) rounds to \(300\) and \(202\) rounds to \(200\). 2. The estimated sum is \(300+200=500\).

Rounded numbers: \(300\) and \(200\). Estimated sum: about \(500\).

- When a three-digit number is close to a whole hundred, use that hundred to judge its size. - Check how far each addend is from a nearby whole hundred. Are those changes small compared with the claimed total?

1. \(398\) rounds to \(400\) and \(210\) rounds to \(200\). 2. The estimate is \(400+200=600\). 3. An answer of \(908\) is far from this estimate, so it is not reasonable.

\(398\) rounds to \(400\) and \(210\) rounds to \(200\). Estimate: \(400+200=600\). No, \(908\) is not reasonable.

- How many groups of about \(3\,\text{m}\) fit into the tower's height? - Use a multiplication fact with tens to check Dylan's guess. - Compare Dylan's predicted height with the given height.

1. Since \(3 \times 20 = 60\), the tower is about \(20\) floors tall. 2. Thirty floors would be about \(30 \times 3 = 90\,\text{m}\), much taller than \(60\,\text{m}\). Dylan's guess is too high.

About \(20\) floors. No, \(30\) floors is too high.

- Round both numbers in each expression to the nearest \(10\). - Write the estimated sum or difference before looking for its match. - Choose the listed exact result closest to that estimate.

1. a) \(392\to390\) and \(184\to180\), so the estimate is \(390+180=570\). The matching exact result is \(576\). 2. b) \(803\to800\) and \(417\to420\), so the estimate is \(800-420=380\). The match is \(386\). 3. c) \(247\to250\) and \(158\to160\), so the estimate is \(250+160=410\). The match is \(405\). 4. d) \(934\to930\) and \(286\to290\), so the estimate is \(930-290=640\). The match is \(648\).

a) Estimate \(570\); match \(576\). b) Estimate \(380\); match \(386\). c) Estimate \(410\); match \(405\). d) Estimate \(640\); match \(648\).

- Round each number to the nearest \(10\) before considering the answer choices. - Write the estimated result as part of your answer. - Choose the option that is close to the estimate and has the right size.

1. a) \(347\to350\) and \(268\to270\), so the estimate is \(620\). The reasonable exact result is \(615\). 2. b) \(823\to820\) and \(376\to380\), so the estimate is \(440\). The reasonable exact result is \(447\). 3. c) \(692\to690\) and \(214\to210\), so the estimate is \(480\). The reasonable exact result is \(478\).

a) Estimate \(620\); choose \(615\). b) Estimate \(440\); choose \(447\). c) Estimate \(480\); choose \(478\).

- Round every original number before doing any arithmetic. - Write the rounded expression so the estimation method can be checked. - Match the result of each rounded expression with one estimate from the list.

1. a) \(289\approx300\) and \(106\approx100\), so \(300+100=400\). 2. b) \(812\approx800\) and \(191\approx200\), so \(800-200=600\). 3. c) \(395\approx400\), \(505\approx500\), and \(99\approx100\), so \(400+500+100=1000\). 4. d) \(725\approx700\) and \(418\approx400\), so \(700-400=300\).

a) \(300+100=400\); estimate \(400\) b) \(800-200=600\); estimate \(600\) c) \(400+500+100=1000\); estimate \(1000\) d) \(700-400=300\); estimate \(300\)

- Find the halfway point between the hundreds on either side of the starting number. - Combine the two jumps before deciding which hundred is nearer to their total. - For the exact landing, use each arrow's full amount, not its rounded amount.

1. The start is \(385\), which rounds to \(400\). The jumps total \(120\), which rounds to \(100\). 2. The estimated landing is \(400+100=500\). 3. The actual jumps land at \(485\) and then \(505\), so the exact landing is \(505\). 4. The exact value is \(5\) greater than the estimate.

Rounded start: \(385\) rounds to \(400\). Rounded total jump: \(100+20=120\), which rounds to \(100\). Estimated landing: \(400+100=500\). Exact landing: \(505\). The exact value is larger.

- Imagine the same number of sections, but with more seats in every section. - Would that new arrangement hold more seats or fewer seats than the hall? - Compare Sam's claim with the larger arrangement.

1. Eight sections of \(70\) seats would hold \(8\times70=560\) seats. 2. Each actual section has fewer than \(70\) seats, so the actual total must be less than \(560\). 3. Sam's \(680\) is greater than \(560\), so it cannot be correct.

\(8\times70=560\). The actual total is less than \(560\), so \(680\) is too large.

- Use a basic multiplication fact for groups of seven, then think in tens. - Compare the exact count with the neighboring hundreds before rounding. - Subtract Clara's estimate from the exact count to find how far off she was.

1. Six groups of seven tens give \(6\times70=420\) apples. 2. The exact total \(420\) rounds to \(400\) to the nearest hundred, matching Clara's estimate. 3. The estimate is \(420-400=20\) apples below the exact total.

Exactly \(420\) apples; rounded to \(400\). Yes, Clara's estimate is reasonable. It is \(20\) apples too low.

- Round each addend to a nearby hundred before estimating. - Find the exact sum separately. - Compare the two totals and consider how the rounding changes affected the estimate.

1. a) The estimate is \(200+300+300=800\). The exact sum is \(198+304+291=793\), so the estimate is greater than the exact value. 2. b) The estimate is \(400+200+100=700\). The exact sum is \(412+189+111=712\), so the estimate is less than the exact value. 3. c) The estimate is \(100+500+200=800\). The exact sum is \(76+513+202=791\), so the estimate is greater than the exact value.

a) Estimate: \(800\); exact: \(793\); estimate is greater. b) Estimate: \(700\); exact: \(712\); estimate is less. c) Estimate: \(800\); exact: \(791\); estimate is greater.

- Round each number to a nearby ten before estimating. - Find the exact value separately after estimating. - Compare the estimate with the exact value and consider how the rounding changes affected the result.

1. a) The estimate is \(780-210-350=220\). The exact difference is \(782-211-349=222\), so the estimate is less than the exact value. 2. b) The estimate is \(940-420-220=300\). The exact difference is \(943-418-222=303\), so the estimate is less than the exact value. 3. c) The estimate is \(570-150-100=320\). The exact difference is \(567-153-104=310\), so the estimate is greater than the exact value.

a) Estimate: \(220\); exact: \(222\); estimate is less. b) Estimate: \(300\); exact: \(303\); estimate is less. c) Estimate: \(320\); exact: \(310\); estimate is greater.

- When rounding to the nearest hundred, use the tens digit to decide whether to round up or down. - Use each estimate to check whether the exact difference is reasonable. - Regroup carefully when finding the exact differences.

1. Estimate: \(400-200=200\). Exact difference: \(402-199=203\). 2. Estimate: \(700-300=400\). Exact difference: \(698-301=397\). 3. Estimate: \(500-300=200\). Exact difference: \(503-275=228\).

1) Estimate: \(200\); exact difference: \(203\) 2) Estimate: \(400\); exact difference: \(397\) 3) Estimate: \(200\); exact difference: \(228\)

- Round both numbers in each difference to the nearest hundred. - Notice whether the three estimates are different or equal. - Calculate the exact differences when the estimates do not distinguish them.

1. For A), the estimate is \(800-300=500\). 2. For B), the estimate is \(900-400=500\). 3. For C), the estimate is \(700-200=500\). 4. Since all three estimates are equal, the estimates alone do not identify the greatest exact difference. 5. The exact differences are A) \(804-297=507\), B) \(901-395=506\), and C) \(702-198=504\). Therefore, A) has the greatest difference.

A) Estimate: \(500\); exact difference: \(507\) B) Estimate: \(500\); exact difference: \(506\) C) Estimate: \(500\); exact difference: \(504\) All three estimates are equal, so the estimates alone do not identify the greatest exact difference. A) has the greatest exact difference.

- Round each class's amount before estimating. - An estimate is close to the exact total but is not the exact total. - Compare the exact total with the goal only after calculating it.

1. Round \(487\) to \(500\) and \(496\) to \(500\), giving an estimate of \(1000\). 2. An estimate of \(1000\) does not prove the exact total is at least \(1000\). 3. The exact total is \(487+496=983\). 4. Since \(983<1000\), the goal was not reached.

a) Estimate: \(1000\) b) No. c) Exact total: \(983\); the goal was not reached.

- Replace each addend with a nearby hundred. - Compare the claimed answer with the size of your estimate. - After judging the claim, calculate the exact sum.

1. Round to the nearest hundred: \(368\approx400\) and \(257\approx300\), so the sum should be about \(700\). 2. \(925\) is far from that estimate, so it is not reasonable. 3. The exact sum is \(368+257=625\).

The answer \(925\) is not reasonable. An estimate is about \(700\), and the exact sum is \(625\).

- One addend is rounded up and the other is rounded down. - Compare the sizes of those two changes. - Use the exact sum to check the direction and size of the estimation error.

1. The exact sum is \(486+237=723\). 2. The estimate \(700\) is less than \(723\). 3. Rounding \(486\) to \(500\) increases it by \(14\), while rounding \(237\) to \(200\) decreases it by \(37\). 4. The decrease is \(23\) greater than the increase, so the estimate is \(23\) less than the exact sum.

a) \(723\) b) The estimate is less than the exact sum. c) The downward rounding change is larger than the upward rounding change, so the estimate is low by \(23\).

- Check each addend separately against the two neighboring hundreds. - Repair the rounding before combining the estimated values. - After finding the exact sum, compare its size with the corrected estimate rather than expecting them to be equal.

1. The incorrect step is \(386\approx300\). To the nearest \(100\), \(386\approx400\). 2. The other rounding is correct: \(247\approx200\). 3. A correct estimate is \(400+200=600\). 4. The exact sum is \(386+247=633\). 5. The exact sum is reasonable because \(633\) is close to the estimate of \(600\).

The incorrect step is \(386\approx300\); it should be \(386\approx400\). A correct estimate is \(600\), the exact sum is \(633\), and \(633\) is reasonably close to \(600\).

- Think about how far apart the two lunch choices are. - A useful estimate for a close decision should keep more place-value detail. - Make both estimates before deciding which one gives better guidance. - Use the exact total only after you have made and compared the estimates.

1. To the nearest \(100\), \(347\approx300\) and \(286\approx300\), so the estimate is \(600\). 2. To the nearest \(10\), \(347\approx350\) and \(286\approx290\), so the estimate is \(640\). 3. The nearest-\(10\) estimate is more useful for choosing between \(600\) and \(650\) lunches because it points to \(650\), while the nearest-\(100\) estimate is too coarse for this close choice. 4. The exact total is \(347+286=633\), so \(650\) lunches are enough and \(600\) are not.

a) \(600\) b) \(640\) c) The nearest-\(10\) estimate is more useful for this decision. d) The exact total is \(633\), so the cafeteria should prepare \(650\) lunches.

- Choose nearby numbers that make both operations easy to do mentally. - Keep each replacement close enough that the estimate still reflects the size of the original expression. - Use the estimate to judge the claimed result before calculating exactly. - Compare the exact value with your estimate at the end.

1. One useful estimate is \(400+200-150=450\). These numbers are close to the original numbers and are easy to combine mentally. 2. The claimed result \(649\) is far larger than an estimate near \(450\), so the claim is not reasonable. 3. The exact value is \(398+204=602\), then \(602-153=449\). 4. The exact value \(449\) is very close to the estimate \(450\), confirming that \(649\) was unreasonable.

One useful estimate is \(450\), so \(649\) is not reasonable. The exact value is \(449\).

- Rounding one addend up and the other down can make the changes partly cancel. - Compare each rounded addend with its original number. Are both changes in the same direction? - One estimate may be below the true sum and another above it. Count the gap on each side.

1. To tens: \(270+210=480\). 2. To hundreds: \(300+200=500\). 3. The exact sum is \(267+214=481\). 4. The tens estimate is \(1\) away and the hundreds estimate is \(19\) away. The tens estimate is closer by \(18\).

Tens estimate: \(480\). Hundreds estimate: \(500\). Exact sum: \(481\). The tens estimate is closer, by \(18\).

- The arrows go backward. To find an earlier number, undo a backward jump. - Combine the three jump sizes before rounding their total to a hundred. - The starting number must be greater than the ending number.

1. The backward jumps move \(120+90+50=260\) in all. 2. Round the ending number \(433\) to \(400\), and the total jump size \(260\) to \(300\). The estimated start is \(400+300=700\). 3. Work backward: \(433+50=483\), \(483+90=573\), and \(573+120=693\). 4. The estimate is \(700-693=7\) above the exact starting number.

Rounded ending number: \(433\) rounds to \(400\). Rounded jump total: \(120+90+50=260\), which rounds to \(300\). Estimated start: \(400+300=700\). Missing numbers in jump order: \(693\), \(573\), \(483\). The estimate is \(7\) too high.

- Use the neighboring hundreds to decide how each chosen number rounds. - Compare each estimated total with \(1000\) by finding the distance between them. - Equal estimates do not always mean equal exact sums.

1. Group A rounds to \(500+400+100=1000\). 2. Group B rounds to \(600+300+100=1000\). 3. Group C rounds to \(300+400+200=900\). 4. Group D rounds to \(500+300+300=1100\). 5. Only A and B have estimates within \(50\) of \(1000\). 6. Their exact sums are A: \(482+408+107=997\) and B: \(591+267+126=984\). 7. A is \(3\) away from \(1000\); B is \(16\) away. Among these two groups, A is closer.

a) A: \(500,400,100\); estimate \(1000\). B: \(600,300,100\); estimate \(1000\). C: \(300,400,200\); estimate \(900\). D: \(500,300,300\); estimate \(1100\). b) A and B. c) A: \(997\); B: \(984\). A is closer to \(1000\) by comparing distances of \(3\) and \(16\).

- Find both estimates before assuming that the more precise rounding must be better. - Compare how each addend changes under each rounding method. - Rounding one number up and another down can make the changes offset each other. - Compare each estimate with the exact sum at the end.

1. To the nearest \(10\), \(105\approx110\) and \(195\approx200\), so the estimate is \(310\). 2. To the nearest \(100\), \(105\approx100\) and \(195\approx200\), so the estimate is \(300\). 3. The exact sum is \(105+195=300\). 4. The nearest-\(100\) estimate is exact. Its rounding changes cancel: one addend decreases by \(5\) and the other increases by \(5\). With nearest-\(10\) rounding, both addends increase by \(5\), making the estimate \(10\) too high.

a) \(310\) b) \(300\) c) \(300\) d) The nearest-\(100\) estimate is closer; it is exact because its two rounding changes cancel.