Math Vale

🌱 Starter Village · Arithmetic

Subtraction

Subtract whole numbers, including borrowing across zeros, and recognise the three stories subtraction tells: take away, compare, and find the missing part.

In short

  • Subtraction measures the gap between two numbers, whether or not anything is taken away.
  • Subtraction and addition undo each other, which gives you a free check every time.
  • Regrouping rewrites the same number in a different form; it never changes its value.
  • A 0 must borrow from its left before it can lend to its right.

Three stories, one operation

Subtraction answers three different-sounding questions with the same calculation:

  • Take away. You had 52 arrows and used 19. How many are left?
  • Compare. Kofi has 52 arrows, Lena has 19. How many more does Kofi have?
  • Missing part. A quiver holds 52. There are 19 in it. How many more fit?

Only the first one physically removes anything. All three are 52 - 19, because subtraction really measures the gap between two numbers.

Subtraction undoes addition

If 19 + 33 = 52, then 52 - 19 = 33 and 52 - 33 = 19. The three numbers form a fact family, and knowing one member gives you the others.

This is also the fastest way to check a subtraction: add your answer to the number you took away. You should land back on the number you started with.

Unlike addition, subtraction is *not* commutative: 52 - 19 and 19 - 52 are different. Order matters.

Regrouping (borrowing)

Work column by column from the ones. When the top digit is smaller than the bottom digit, you cannot subtract yet — you regroup.

Take 1 from the column to the left. It is worth 10 in the column you are in.

52 - 19 ---- Ones: 2 - 9 does not work. Regroup: the 5 tens become 4 tens, and the 2 ones become 12 ones. Ones: 12 - 9 = 3. Tens: 4 - 1 = 3. Result: 33.

Nothing has changed about the number 52 — it has only been rewritten as 4 tens and 12 ones.

Borrowing across zeros

A 0 has nothing to give away, so it borrows first and then passes the value on.

500 - 246 ----- Ones: 0 - 6 does not work, and the tens digit is 0 too. Go to the hundreds: 5 hundreds become 4 hundreds and 10 tens. Now the tens can lend: 10 tens become 9 tens and 10 ones. Ones: 10 - 6 = 4. Tens: 9 - 4 = 5. Hundreds: 4 - 2 = 2. Result: 254.

If borrowing across zeros feels fiddly, count up instead: from 246 to 250 is 4, from 250 to 500 is 250, giving 254.

Choosing subtraction in a word problem

Words that often signal subtraction: *left, remaining, fewer, how many more, difference, decrease, spent*.

"How many more" is the one students most often get wrong, because nothing is taken away. It is still subtraction: it asks for the size of the gap.

Always sanity-check the answer against the story. What is left can never be more than you started with.

Worked examples

Example 1

4,003 - 1,568 = ?

  1. Estimate: about 4,000 - 1,600 = 2,400.
  2. Ones: 3 - 8 does not work. The tens and hundreds are both 0, so borrow from the thousands.
  3. 4 thousands become 3 thousands, 10 hundreds; then 9 hundreds, 10 tens; then 9 tens, 13 ones.
  4. Ones: 13 - 8 = 5. Tens: 9 - 6 = 3. Hundreds: 9 - 5 = 4. Thousands: 3 - 1 = 2.
  5. 4,003 - 1,568 = 2,435. Check: 2,435 + 1,568 = 4,003.

Example 2

Tariq has 312 gold coins and Noor has 178. How many more coins does Tariq have than Noor?

  1. Nothing is spent — the two amounts are being compared — but comparison is still subtraction.
  2. Number sentence: 312 - 178.
  3. Ones: 2 - 8 needs a regroup: 12 - 8 = 4.
  4. Tens: 0 - 7 needs a regroup: 10 - 7 = 3. Hundreds: 2 - 1 = 1.
  5. Tariq has 134 more coins. Check: 178 + 134 = 312.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

15 - 3 = ?

Answer: 12

  1. 15 - 3
  2. Subtract each column: ones, then tens, then the rest.
  3. = 12
  4. Check: 12 + 3 = 15

Problem 2

Difficulty 3 of 5

717 - 27 = ?

Answer: 690

  1. 717 - 27
  2. Regroup from the left so every column can be subtracted.
  3. = 690
  4. Check: 690 + 27 = 717

Problem 3

Difficulty 4 of 5

? - 854 = 5,992

Answer: 6846

  1. whole - part = part, so whole = part + part.
  2. 854 + 5,992 = 6,846
  3. Check: 6,846 - 854 = 5,992

Common mistakes

  • Taking the smaller digit from the larger one in each column regardless of which is on top: 52 - 19 written as 47.
  • Forgetting to reduce the digit that was borrowed from.
  • Getting lost borrowing across zeros and skipping a column.
  • Adding instead of subtracting on a "how many more" question.

What you should be able to do

  • Recall subtraction facts within 20 automatically.
  • Subtract multi-digit numbers with regrouping, including across zeros.
  • Use addition to check a subtraction result.
  • Choose subtraction for "how many are left" and "how many more" problems.

Where this fits in the curriculum

Common Core

  • 2.OA.B.2

    Grade 2 — Fluently add and subtract within 20 from memory.

  • 4.NBT.B.4

    Grade 4 — Fluently add and subtract multi-digit whole numbers using the standard algorithm.

  • 2.OA.A.1

    Grade 2 — Use addition and subtraction within 100 to solve take-from, compare and unknown-addend problems.

    This is where "the three stories subtraction tells" is named: taking away, comparing and finding a missing part are one standard.

Ontario

  • G2.B2.2

    Grade 2 — Recall and demonstrate subtraction facts for numbers up to 20, alongside the related addition facts.

    Ontario deliberately teaches addition and subtraction as one expectation rather than two, which is why this code and B2.4 are shared with the addition skill.

  • G3.B2.4

    Grade 3 — Demonstrate an understanding of algorithms for adding and subtracting whole numbers.

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