Subtracting three-digit numbers by place value, including across zero

When you reach for it

Any time a three-digit amount is taken away, compared with another, or split into a known part and an unknown part, the question is a subtraction within 1{,}000. A number such as 642 is 6 hundreds, 4 tens and 2 ones (Reading and writing numbers to 1,000), which Expanded form writes as 600 + 40 + 2. To take 278 away from it, take hundreds from hundreds, tens from tens and ones from ones. When a place does not hold enough, trade: one ten for ten ones, exactly as in Regrouping in subtraction, or, new here, one hundred for ten tens. When the place you need to trade from is empty, trade twice in a row.

One picture carries the whole lesson. At a pumpkin farm, Wanjiku runs the apple barn. Apples are packed in crates of 100, bags of 10, and a few loose apples. Opening a crate gives ten bags. Opening a bag gives ten loose apples. Neither move changes how many apples there are, and every trade in this lesson is one of those two moves. Base ten blocks say the same thing: a flat is a hundred, a rod is a ten, a unit is a one, and a flat can be swapped for ten rods or a rod for ten units.

The lesson runs over four days. Day 1 takes away with no trade. Day 2 trades a ten, a hundred, or both. Day 3 meets a zero in the tens place. Day 4 meets two zeros and checks every answer by adding back. Each day moves from blocks to quick drawings (a square for a hundred, a stick for a ten, a dot for a one) to writing by place. The column method that crosses out digits comes later, in the standard subtraction algorithm; here every trade is shown with pieces or written as hundreds, tens and ones, so the child can see what each trade means.

The procedure

Day 1: hundreds from hundreds, tens from tens, ones from ones

Wanjiku has 758 apples: 7 crates, 5 bags and 8 loose. A store buys 325 apples, which is 3 crates, 2 bags and 5 loose. Build 758 with blocks: 7 flats, 5 rods and 8 units. Then take away 3 flats, 2 rods and 5 units.

Count what is left by kind. Flats: 7 - 3 = 4. Rods: 5 - 2 = 3. Units: 8 - 5 = 3. That is 4 flats, 3 rods and 3 units, so

Written in expanded form, the same work is

and 400 + 30 + 3 = 433. A place value table keeps the columns straight. Each column is one kind of block.

Why taking place by place is allowed. Taking 325 away means taking away its pieces: 3 hundreds, 2 tens and 5 ones. Taking them one kind at a time removes exactly the same apples as taking them all at once, so the amount left is the same. And 7 hundreds take away 3 hundreds is 4 hundreds for the same reason that 7 - 3 = 4: a hundred is one thing, a crate, and crates are counted like anything else. That is also why the places must match. The store asked for 3 crates; taking 3 loose apples instead would take far too few.

On Day 1 every place of the first number holds at least as much as the same place of the second, so no trade is needed. The child may start at the hundreds or at the ones; when nothing is short, the order does not matter.

Day 2: trading a ten, then a hundred

Now the order from the question: 642 - 278. Wanjiku has 6 crates, 4 bags and 2 loose apples. The store wants 2 crates, 7 bags and 8 loose. Look at each place before taking anything. The ones are short: 2 loose and 8 wanted. The tens are short: 4 bags and 7 wanted. The hundreds have plenty.

Start with the ones. Open one bag: now there are 6 crates, 3 bags and 12 loose apples, still 600 + 30 + 12 = 642. Then the tens: 3 bags and 7 wanted. Open one crate: now there are 5 crates, 13 bags and 12 loose apples, still 500 + 130 + 12 = 642. Every place now has enough.

Ones: 12 - 8 = 4. Tens: 13 - 7 = 6. Hundreds: 5 - 2 = 3.

Wanjiku hands over 2 crates, 7 bags and 8 loose apples, and keeps 3 crates, 6 bags and 4 loose.

Why a trade is allowed. A hundred is ten tens and a ten is ten ones. Opening a crate changes how the apples are packed, not how many there are. So “6 hundreds, 3 tens, 12 ones” and “5 hundreds, 13 tens, 12 ones” are both names for 642, written in pieces that make the taking away possible. A place may hold more than nine for a while. That is a working form, not a new number.

Why start with the ones. Trading for the ones takes a ten out of the tens place, so the tens change before anyone looks at them. Working up from the ones settles each place once. Starting with the tens can mean going back: in 652 - 258 the tens look fine (5 and 5), but the ones need a ten, which leaves 4 tens, and then the tens need a hundred after all. So the place to compare is the one after any trade below it.

Day 3: a zero in the tens

705 - 268. Wanjiku has 7 crates and 5 loose apples and no bags at all. The order is 2 crates, 6 bags and 8 loose. The ones are short, 5 and 8 wanted, so a bag must be opened. But there is no bag to open.

So open a crate first. That gives 6 crates, 10 bags and 5 loose. Now there are bags, so open one of them: 6 crates, 9 bags and 15 loose. That is still 600 + 90 + 15 = 705.

Why it works. These are the same two trades as on Day 2, and each keeps the amount the same. The zero only fixes their order: a ten cannot come out of an empty place, so the hundred has to become tens first. After trading across a zero the tens place always holds 9, the 10 new tens less the one that went to the ones.

A zero in the ones place is nothing new. It is a short ones place like any other, and the tens place has a ten to give, as on Day 2.

Day 4: two zeros, and checking by adding back

800 - 356. Eight crates, no bags, no loose apples. The ones are short and the tens are empty, so the trades run as on Day 3. Open a crate: 7 hundreds, 10 tens, 0 ones. Open a bag: 7 hundreds, 9 tens, 10 ones. Now take away 3 hundreds, 5 tens and 6 ones.

Ones: 10 - 6 = 4. Tens: 9 - 5 = 4. Hundreds: 7 - 3 = 4.

Checking by adding back. The apples handed over and the apples kept make up all the apples Wanjiku started with. So 444 + 356 must come back to 800. Add by place, as in the partial sums method:

since 400 + 300 = 700, 40 + 50 = 90 and 4 + 6 = 10. Look at how the pieces close up. The 10 ones make a new ten, and with the 9 tens that is 10 tens, a new hundred, and with the 7 hundreds that is 8. The trades of the subtraction come back in reverse: the bag that was opened is filled again, and so is the crate. This is the parts and whole idea from number bonds at a bigger size: the part taken and the part left make the whole, which is also why a subtraction can be read as a missing part.

A second check is to count up from 356: 44 more reaches 400 and 400 more reaches 800, so the difference is 44 + 400 = 444.

Worked examples

Examples 1 and 2 belong to Day 1, 3 and 4 to Day 2, 5 and 6 to Day 3, and 7 and 8 to Day 4. Petar and Kainoa belong to a train set club that keeps its track pieces in boxes of 100, packs of 10 and single pieces.

Boundaries

• The first number must be at least as large as the second. 325 - 758 has no answer among the whole numbers, since there is nowhere above the hundreds to trade from. When the first number is the larger one, the hundreds are never short once the ones and tens are settled (Exercise 22).
• A trade is one piece for ten of the next smaller kind. A hundred becomes ten tens and a ten becomes ten ones. A hundred is also 100 ones, but nobody draws 100 dots; across a zero the two trades in a row do the same job in two small steps.
• A trade changes the pieces, never the amount. Every row of a quick drawing before the crossing out is the first number. A place may hold 10 to 19 while working, but the amount left always has a single digit in each place, because a place is only traded into when it is short: it then holds less than ten more than what is taken from it.
• Within 1{,}000. The largest first number here is 1{,}000, ten hundreds. Past that the numbers have a thousands place, and the same trades continue one place further up.
• Like from like. Hundreds are taken only from hundreds, tens from tens and ones from ones. A digit means nothing on its own; it counts crates, bags or loose apples according to its place.

Where this goes next

The same trades, recorded by crossing out a digit and writing a smaller one above it, are the standard subtraction algorithm. Zeros in longer numbers, and a shortcut that reads 800 as 80 tens, are the subject of subtracting across zeros. The adding companion is the partial sums method, which is also the tool for checking by adding back. Counting up gives a second way to find a difference, often quicker when the first number has zeros. Three-digit differences then turn up in story problems of every kind: taking away, comparing, and finding a missing part.

Exercises

Part 1: no trade

1. A pile has 6 flats, 8 rods and 4 units. Take away 3 flats, 1 rod and 2 units. Write the subtraction and find what is left.
2. Find 896 - 534.
3. Find 475 - 205.
4. Wanjiku has 968 apples and sells 520. How many apples are left?
5. Write 759 and 413 in expanded form. Use them to find 759 - 413.

Part 2: trading a ten, a hundred, or both

6. Look at each place first. Which place needs a trade in (a) 563 - 238, (b) 563 - 381, (c) 563 - 297? Then find all three differences.
7. Find 734 - 186. Write each trade in a place value table.
8. Find 451 - 175.
9. A quick drawing shows 3 squares, 15 sticks and 6 dots. What number is it?
10. Petar has 825 track pieces and Kainoa has 368. How many more pieces does Petar have?
11. Find 382 - 145 and check by adding back.

Part 3: a zero in the first number

12. Find 506 - 234.
13. Find 302 - 127.
14. Find 608 - 459. How many ones are there after both trades?
15. Wanjiku has 4 crates and 7 loose apples. She sells 158 apples. How many are left?
16. Find 750 - 316.

Part 4: two zeros, and checking

17. Find 500 - 263 and check by adding back.
18. Find 900 - 418 and check by adding back.
19. Find 700 - 9.
20. The club owns 1{,}000 track pieces, and 572 are on the layout. How many pieces are in the boxes?
21. Find the missing number: 600 - \square = 275.

Reasoning

22. The first number is larger than the second. Can the hundreds place still be short after the ones and tens have been traded? Explain.
23. Show that 9 hundreds 0 tens 2 ones, 8 hundreds 10 tens 2 ones, and 8 hundreds 9 tens 12 ones are all the same number.
24. Find two subtractions of three-digit numbers with a difference of 248: one that needs no trade and one that needs two trades.
25. A three-digit number ends in two zeros, and 1 is taken away from it. Why does the answer always end in 99?