Addition and Subtraction of Whole Numbers to 1,000,000
Warm-up
Quick mental math using known facts: 7+8=15, so 70+80=? (150). 700+800=? (1,500). 7,000+8,000=? (15,000). 70,000+80,000=? (150,000). Pattern: the sum digit is always 15, with an increasing number of zeros. This is the annex-zeros strategy applied as a pattern.
Explore
Community fundraising challenge: three towns raised funds for a shared project. Town A: 289,701. Town C: 600,000. How far from the goal? Students solve using two strategies each, show an estimate first.
Formalize
Front-end estimation comparison: estimate 467,329 + 321,897 using (1) front-end (400,000+300,000=700,000) and (2) rounding to nearest 10,000 (470,000+320,000=790,000). Actual: 789,226. Which estimate was closer? (The rounded estimate.) When is front-end enough? (When precision is not needed.)
Addition and Subtraction of Whole Numbers to 1,000,000
Inverse verification: computed 823,400 - 256,178 = 567,222. Check: 567,222 + 256,178 = 823,400. Work it through. Consistent verification at this scale builds accuracy and metacognitive awareness.
Practice
Students solve 6 six-digit computation problems with estimates and strategies shown. Exit ticket: use annex zeros to solve 80,000 + 70,000 mentally.
Exit ticket
Students solve 6 six-digit computation problems with estimates and strategies shown. Exit ticket: use annex zeros to solve 80,000 + 70,000 mentally.
Step 1: Estimate: 178,450 ≈ 180,000; 96,875 ≈ 100,000 → about 280,000. Note the estimate leans HIGH (both roundings went up) — expect the true sum a bit under.
Step 2: Compute with the algorithm, narrating trades: ones 0+5=5; tens 5+7=12 → 2 carry 1; hundreds 4+8+1=13 → 3 carry 1; thousands 8+6+1=15 → 5 carry 1; ten-thousands 7+9+1=17 → 7 carry 1; hundred-thousands 1+0+1=2. Sum: 275,325.
Step 3: Reconcile with the estimate: 275,325 vs 280,000 — under, as predicted, by about the amount the roundings added (≈4,700). When the error direction was predictable, checking becomes surgical, not vague.
Step 4: The place-value slip to guard: numbers of DIFFERENT lengths (6-digit + 5-digit) must right-align. Write 96,875 with its ten-thousands under the 7, not under the 1. One column of misalignment produces garbage that only the estimate will catch.
Weapon 1 — count up (clean): 268,314 + 1,686 → 270,000; + 30,000 → 300,000; + 200,000 → 500,000. Total: 1,686 + 30,000 + 200,000 = 231,686.
Weapon 2 — shift both down 1: 499,999 − 268,313. Every column subtracts without borrowing: 231,686. ✓ agreement.
Weapon 3 — the algorithm: borrowing across FIVE zeros — a cascade where 500,000 becomes 4-9-9-9-9-10 territory. It works, but each 9 is a chance to slip.
Step 4: Debrief the choice: subtrahend full of zeros → count up or shift; mixed digits like 743,251 − 268,314 → algorithm is fine. The meta-skill (diagnose, THEN compute) is worth more than any single method — say it every time until they roll their eyes.
Reality anchor: 500,000 seats, 268,314 tickets sold → 231,686 seats left. Numbers this size are stadium-sized; name the context to keep them from floating.
The story: the city library holds 482,600 items. This year it added 27,850 new items and discarded 15,375 worn ones. The branch across town holds 316,900 items after its own updates. How many more items does the main library now hold?
Step 1: Plan the pipeline before arithmetic: update the main count (add, then subtract), THEN compare to the branch.
Step 2: Update: 482,600 + 27,850 = 510,450. Then 510,450 − 15,375 = 495,075.
Step 3: Compare: 495,075 − 316,900. Count up: 316,900 + 83,100 → 400,000; + 95,075 → 495,075. Difference: 178,175.
Step 4: Estimate-audit the whole chain at once: ≈ 483 + 28 − 15 = 496 thousand; minus 317 thousand ≈ 179 thousand ✓.
Step 5: Answer the question ASKED, in a sentence: "The main library holds 178,175 more items." Multi-step problems are lost at the plan stage or the last-sentence stage far more often than in the arithmetic — grade all three stages.