Capstone — Choosing Your Tool

Four lessons, four tools: completing the ten, completing the hundred and compensation, column addition, column subtraction. None of them is "the" way to add or subtract — each one wins on a different shape of problem. The real skill this capstone is about isn't doing any one of them faster. It's noticing, before you start, which one a problem is asking for.

See it first

The habit behind completing the ten and completing the hundred is really one habit: find the gap to the nearest round landmark, whatever size that landmark is.

Reading a problem before solving it

Here's the question to ask before you commit to a method: is a round number close by, or not?

• Close by → completing the ten/hundred, or compensation. Fast, and it stays fast how far away it looks — one detour to the round number, one correction.
• Not close by, and several digits → column addition or subtraction. Slower on easy problems, but it never loses a digit, however awkward the numbers are.

Compensation works for subtraction too, but the correction runs the other way: you take off the round amount, then give back the shortfall instead of paying it.

Two more things worth noticing before you commit to a method: a four-digit number simply has more columns than a two-digit one — more places a mental running total could lose track — which is exactly the situation column addition is built for. And whenever a column's top digit is smaller than the one you're subtracting, that column has to borrow — sometimes cascading through several zeros in a row, exactly like 604 - 258 did in the subtraction lesson.

Estimate first, always

Whichever tool you reach for, one habit protects you from every kind of mistake: before you compute, guess roughly. 4,867 + 2,385 is a bit under 5,000 + 2,400, so the real answer should land somewhere close to 7,400 — if column addition gives you something wildly different, you know to recheck a column, not to trust the wrong answer just because it came from the "right" method.

❓ Cross-question — "Doesn't estimating first take extra time?" A rough estimate takes seconds and catches almost every slip — a dropped carry, a misread digit, a compensation done backward. The time it saves when it catches a mistake is much larger than the time it costs when it doesn't.

🎯 Your turn

Look at 5,986 + 24 and 3,412 - 1,687 before doing either one. For each, decide out loud which tool you'd reach for and why — then solve it that way. Bring that reasoning into the exercise below.