Free math worksheets

- Compare \(80\,\text{cm}\) with your own height. - Think of an object that is about \(80\,\text{cm}\) long. - Could an entire class and its furniture fit in a space that length? - Might the unit have been recorded incorrectly?

1. Since \(100\,\text{cm} = 1\,\text{m}\), \(80\,\text{cm}\) is less than \(1\,\text{m}\). 2. A classroom must be long enough to hold many students, desks, and chairs. 3. Therefore, \(80\,\text{cm}\) is far too short to be the length of a classroom.

The measurement is not reasonable because \(80\,\text{cm}\) is less than \(1\,\text{m}\), while a classroom is several meters long.

- 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 \(300\)

- 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 \(449\).

- Round each selected number to the nearest \(100\) before doing any exact addition. - Compare each estimated sum with \(1000\) using the distance from \(1000\). - Exact addition is needed only after the estimation stage identifies the candidate groups.

1. Group A: \(482\to500\), \(408\to400\), \(107\to100\), so the estimate is \(1000\). 2. Group B: \(591\to600\), \(267\to300\), \(126\to100\), so the estimate is \(1000\). 3. Group C: \(334\to300\), \(408\to400\), \(215\to200\), so the estimate is \(900\). 4. Group D: \(482\to500\), \(334\to300\), \(267\to300\), so the estimate is \(1100\). 5. Only Groups 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. Their distances from \(1000\) are \(3\) and \(16\), so Group A is closest.

A: estimate \(1000\); exact sum \(997\) B: estimate \(1000\); exact sum \(984\) C: estimate \(900\) D: estimate \(1100\) Only A and B are within \(50\) of \(1000\). Group A is actually closest, with \(997\).

- 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.