Add and subtract within one thousand with explained exchanges

Learning goal: Add and subtract within 1,000 using hundreds, tens and ones, record a written method and explain exchanges among units. Start with models or drawings, then connect them with equations. The goal is reasoning about place value, not an unexplained memorized column procedure.

For 268 + 157, combine corresponding units: two hundreds plus one hundred, six tens plus five tens, and eight ones plus seven ones. This gives three hundreds, eleven tens and fifteen ones. Each count names its unit; adding 2 + 1 alone would lose the hundreds' value.

Compose ten of the fifteen ones into one ten. Now there are three hundreds, twelve tens and five ones. Compose ten of the twelve tens into one hundred. The final representation is four hundreds, two tens and five ones: 425. The exchanges preserve the amount because ten ones equal one ten and ten tens equal one hundred.

A partial-sums method records the same reasoning differently: 200 + 100 = 300, 60 + 50 = 110, and 8 + 7 = 15. Add 300 + 110 + 15 = 425. Explain how each partial sum corresponds to a place in the original addends. The order of regrouping may vary while the final value remains the same.

For subtraction, consider 402 − 178. The usual starting representation has four hundreds, zero tens and two ones. To remove eight ones, first exchange one hundred for ten tens, producing three hundreds, ten tens and two ones. Then exchange one of those tens for ten ones, producing three hundreds, nine tens and twelve ones.

Now subtract the matching units: twelve ones minus eight ones leaves four; nine tens minus seven tens leaves two tens; three hundreds minus one hundred leaves two hundreds. The result is 224. The zero tens did not prevent subtraction; it required a meaningful exchange from a larger unit.

The illustration lists these equivalent forms of 402 before subtraction. It does not count the rows as separate piles. In particular, 3 hundreds + 9 tens + 12 ones is 300 + 90 + 12 = 402. Writing 3, 9 and 12 as if they were ordinary numeral digits would not make a valid hundreds-tens-ones numeral.

Use the inverse relationship to check: 224 + 178 = 402. Also estimate the result's size informally. Removing a little less than 200 from a little more than 400 should leave a little more than 200, consistent with 224. An estimate helps detect a large error but does not replace the exact calculation and explanation.

Worked example 1: Compose in addition. From three hundreds, eleven tens and fifteen ones, exchange ten ones first and then ten tens. The final four hundreds, two tens and five ones name 425. Every exchange has an equality behind it.

Worked example 2: Decompose across a zero. Rewrite 402 as 3 hundreds + 10 tens + 2 ones, then as 3 hundreds + 9 tens + 12 ones. Check 300 + 90 + 12 = 402 before removing 178.

Worked example 3: Verify a difference. After finding 224, add 178. Eight ones and four ones make twelve, seven tens and two tens plus the composed ten make ten tens, and the hundreds combine to recover 402. The check confirms the total-parts relationship.

Guided practice: Model additions requiring an exchange of ones, tens or both. Model subtraction with and without a zero in the tens place. Have learners explain every rewritten representation before subtracting, so an exchange is not mistaken for changing the starting amount.

Independent tasks: Solve one unfamiliar three-digit addition and subtraction within the range using a drawing and numerical record. Explain a composed or decomposed unit and check by the inverse operation. Offer another example later to distinguish transferable reasoning from copying the model's digits.

Feedback: If an exchange is described only as borrowing, ask what unit was exchanged and why its value is unchanged. If the learner subtracts the smaller digit from the larger regardless of position, return to the original amount and actual amount removed. Require consistent place-value units.