Round using bounding multiples and midpoint distance

Learning goal: Round numbers to a named place from millions through hundredths by locating the neighboring multiples and deciding which is nearer. Rounding replaces an exact value with a nearby value useful for a purpose. The requested place determines the size of the interval.

To round 6.347 to the nearest hundredth, locate 6.34 and 6.35. These consecutive hundredths differ by 0.01. Their midpoint is 6.345. Since 6.347 is above the midpoint, it is closer to 6.35. Its distance from 6.34 is 0.007 and from 6.35 is 0.003. Round to 6.35. The number line gives meaning to the familiar decision based on the next digit.

Round the same number to the nearest tenth. Now the neighbors are 6.3 and 6.4, with midpoint 6.35. The original 6.347 is below 6.35, so it rounds to 6.3. The hundredth-place answer and tenth-place answer differ because their intervals differ. Write the target place before calculating so you do not answer a different question.

In this lesson, if a positive number is exactly halfway between the neighboring multiples, round to the larger one. Thus 6.345 rounds to 6.35 to the nearest hundredth. State the convention when discussing a midpoint; the location is equally far from both neighbors, so distance alone does not break the tie.

Crossing a place-value boundary may require regrouping. The neighbors of 9.996 to the nearest hundredth are 9.99 and 10.00. The midpoint is 9.995, and 9.996 is above it, so the rounded value is 10.00. The two decimal zeroes communicate the hundredth place used for rounding. They do not make the numerical value different from ten.

Whole-number places use the same reasoning. For 2,746,381.62 to the nearest million, the bounding multiples are 2,000,000 and 3,000,000. The midpoint is 2,500,000, so the result is 3,000,000. To the nearest hundred thousand, the neighbors are 2,700,000 and 2,800,000, and the midpoint is 2,750,000. The original is below that midpoint, giving 2,700,000.

For that same original value, nearest ten thousand gives 2,750,000; nearest thousand gives 2,746,000; nearest hundred gives 2,746,400; nearest ten gives 2,746,380; and nearest one gives 2,746,382. Each answer is a multiple of the requested unit. Rounding to a larger place creates a coarser estimate, which may be suitable for a quick population-scale comparison but unsuitable for paying an exact bill.

Always round from the original number unless a task explicitly asks something else. Rounding 6.347 first to hundredths gives 6.35, then rounding that result to tenths gives 6.4. Directly rounding the original to tenths gives 6.3. Repeated rounding can change the answer because the first step discards information about which side of the later midpoint the original occupied.

Worked decision: A length is 3.684 meters. Nearest hundredth is 3.68 because the original is below 3.685, halfway between 3.68 and 3.69. Nearest tenth is 3.7 because the original exceeds 3.65, halfway between 3.6 and 3.7.

Worked error check: A student gives 2,746,380 for rounding to the nearest hundred. That result is not a multiple of 100. The correct neighboring hundreds are 2,746,300 and 2,746,400, and the latter is nearer.

Guided practice: For each target place, write lower neighbor, midpoint, upper neighbor and original value. Sketch a line with equal intervals and explain the decision. Include an exact midpoint and a result crossing a whole-number boundary.

Independent performance: Round one large decimal to a large whole-number place, to ones, to tenths and to hundredths. Explain why the purpose determines the useful precision. Diagnose a double-rounding error and retain the original value for every direct decision. Evaluate the stated target, chosen bounds and reasoning separately from copying a digit rule.