Use place value to round, add and subtract

Learning goal: Use place value to round to tens and hundreds, calculate within one thousand and multiply one-digit numbers by multiples of ten. A digit's value depends on its position. In 347, the three represents three hundreds, the four represents four tens and the seven represents seven ones. These are different-size units within one number.

To round 347 to the nearest ten, locate the neighboring multiples of ten: 340 and 350. Their midpoint is 345. Since 347 is above the midpoint, it is closer to 350. The distances confirm this: seven from 340 and three from 350. At an exact halfway point such as 345, this whole-number rounding convention chooses the greater neighboring ten, 350.

To round the same number to the nearest hundred, use 300 and 400, whose midpoint is 350. Now 347 is below the midpoint and closer to 300. The requested place changes the answer. Rounding is not deleting digits; it replaces a number with a nearby multiple of the requested unit. Identify the endpoints before deciding.

Addition can combine corresponding units. For 376 + 248, add hundreds, tens and ones: 500 + 110 + 14 = 624. Regroup fourteen ones as one ten and four ones. Together with the other tens, there are twelve tens; regroup ten of them as one hundred. The result contains six hundreds, two tens and four ones. Every regrouping preserves the total value. A written column record shows the same steps in less space: add the ones first and write the new ten above the tens column, then add the tens and write the new hundred above the hundreds column.

Subtraction can rename units when a place does not contain enough. For 503 − 267, rename one hundred as ten tens, then rename one ten as ten ones. Five hundred three becomes four hundreds, nine tens and thirteen ones. Subtract two hundreds, six tens and seven ones: two hundreds, three tens and six ones, or 236. The zero in the original tens place does not mean subtraction is impossible.

An addition check supports subtraction: 236 + 267 = 503. Estimate too: 500 − 270 = 230, so 236 is reasonable. Neither an estimate nor a successful check explains the regrouping by itself. Use both reasoning and verification to build efficient, accurate calculation.

Multiplying tens uses the same unit structure. For 6 × 40, think six groups of four tens. Six times four is twenty-four, so the result is twenty-four tens, or 240. A drawing of six groups of four ten-rods would show 240 individual units. Saying “add a zero” may produce this answer, but counting tens explains why it works for these whole-number factors.

Worked example: Find 583 + 246. Combine 500 + 200 = 700, 80 + 40 = 120, and 3 + 6 = 9. The sum is 829. Rename twelve tens as one hundred and two tens if writing the result in standard place-value form. Estimate 580 + 250 = 830, close to the exact sum.

Worked rounding example: Round 650 to the nearest hundred. Its neighbors are 600 and 700, and it lies exactly halfway. Under the rounding convention used here, choose 700. To the nearest ten, 650 remains 650 because it is already a multiple of ten.

Worked multiplication: Seven groups of eighty are seven groups of eight tens. There are fifty-six tens, so 7 × 80 = 560.

Practice with representations: Use base-ten drawings or blocks to explain one addition with regrouping and one subtraction across a zero. Compare two valid methods, such as renaming units and counting up from the smaller number. Then practice until calculations become accurate and efficient without requiring a full block drawing every time.

Independent evidence: Round numbers near and at midpoints, calculate unfamiliar sums and differences within one thousand, and explain several products involving tens. Always state the rounding place and check that the chosen multiple belongs to that place. If a regrouping changes the value, compare the represented quantity before and after the trade.