Regrouping in Subtraction

In this lesson I show how to teach regrouping (borrowing) in subtraction. We start out by teaching the actual regrouping separately, and then transfer that step-by-step into the subtraction algorithm.

First, teach the students about regrouping as explained in the box below.

As a first step, we study breaking a ten-pillar into ten little cubes. This is also called “regrouping”, because one ten “changes groups” from the tens group into the ones.   Break a ten. 4 tens 5 ones 3 tens 15 ones First we have 45. We “break” one ten-pillar into little cubes. Now we have 3 tens and 15 ones. It is still 45, but written in a different way.

Here is another example. First we have 5 tens 3 ones. We “break” one ten-pillar into 10 little cubes. We end up with 4 tens 13 ones.   Break a ten. 5 tens 3 ones 4 tens 13 ones

Example exercises follow. Here, students practice regrouping by itself, without sutbracting anything.

Break a ten into 10 ones. What do you get? You can draw ten-pillars and cubes to help.

Break a ten.   a.  3 tens 0 ones   2 tens ___ ones

Break a ten.   b.  __tens __ones   __tens __ones

Break a ten.   c.  __tens __ones   __tens __ones

Break a ten.   d.  __tens __ones   __tens __ones

Let's study subtraction. The pictures on the right illustrate 45 – 17. First, a ten is broken into 10 ones. So, 4 tens 5 ones becomes 3 tens 15 ones. After that, cross out (subtract) 1 ten 7 ones. Break a ten. 4 tens 5 ones 3 tens 15 ones Cross out 1 ten 7 ones from the secondpicture. What is left? ___ tens ___ ones

The pictures on the right illustrate 52 – 39. First, a ten is broken into 10 ones. So, 5 tens 2 ones becomes 4 tens 12 ones. After that, cross out (subtract) 3 tens 9 ones. Break a ten. 5 tens 2 ones 4 tens 12 ones Cross out 3 tens 9 ones from the secondpicture. What is left? ___ tens ___ ones

  In the following exercises, the regrouping is done in the pictures. Students take note of that, then "cross out" or subtract something.

Break a ten. 3 tens 6 ones 2 tens 16 ones a.  Cross out 8 ones from the secondpicture. What is left? ___ tens ___ ones

Break a ten. ___ tens ___ ones ___ tens ___ ones b.  Cross out 2 tens 7 ones from the secondpicture. What is left? ___ tens ___ ones

In these exercises, students regroup, then subtract. The presentation uses "tens" and ones.

a.  5 tens 5 ones – 1 ten  7 ones 4 tens 15 ones – 1 ten    7 ones 3 tens   8 ones

b.  7 tens 2 ones –  3 tens 5 ones  __ tens __ ones –  3 tens   5 ones __ tens __ ones

g.  8 tens 1 one   –  6 tens 5 ones  __ tens __ ones –  6 tens 5 ones __ tens __ ones

h.  6 tens 3 ones –  2 tens 8 ones  __ tens __ ones –  2 tens 8 ones __ tens __ ones

After this, the students should be ready to learn the usual form of subtracting where the numbers are in columns.

The picture illustrates subtracting 16 from 53. First, we break a ten into ten ones. Then we cross out 1 ten 6 ones. When the subtraction is written down in columns, we cross the “5” in the tens-column and write 4 above it. We also cross the “3” in the ones column and write 13 above it. This shows the same thing as the pictures: one of the tens is "broken down" into ten ones, so there is one less ten in the tens column, and 10 more ones in the ones column. Then we can subtract tens and ones separately.

5 tens, 3 ones 4 tens, 13 ones Cross out 1 ten 6 ones from thesecondpicture. What is left? ___ tens ___ ones 4 13 5 3 – 1 6 3 7

Now students can practice the normal way of subtraction in columns. The exercises below have the additional prompt for students to regroup before they subtract in columns.

a.  6 tens 3 ones  →  5 tens  13 ones Take away 1 ten, 7 ones. 5 13 6 3 – 1 7

c.  6 tens 0 ones →  ___ tens ___ ones Take away 3 tens, 9 ones.   6 0 – 3 9

e.  3 tens, 5 ones  → ___ tens ___ ones Take away 1 ten, 7 ones.   3 5 – 1 7

g.  7 tens, 6 ones  → ___ tens ___ ones Take away 4 tens, 8 ones.   7 6 – 4 8

When students do normal subtraction problems, instruct them to always check their answer by adding.

Check:   4  16 1 a. 5 6 –  2 7 2 9 2 9 +  2 7 5 6	Check:     b. 9 0 –  2 8    +  2 8
d. 9 0 –  3 5	e. 8 2 –  2 5
j. 5 5 –  1 7	k. 3 1 –  1 8