Adding Two-Digit Numbers: Composing a Ten (Regrouping) - First Grade Mathematics

Welcome to one of the most exciting breakthroughs in first grade mathematics: composing a ten, also known as regrouping! What happens when you add two numbers like 28 + 15, and the ones add up to more than 9? In our number system, you can NEVER write two digits in a single place-value column. If your ones pile reaches 10 or more, you must gather ten loose units and bundle them into a brand-new rod of ten! In this chapter, you will discover the magic of composing a ten and learn how to carry that new ten into the tens column.

The Problem: The Ones Column Overflows!

Look at what happens when we try to add 28 + 15:

Adding 28 + 15:
Look at the Ones: 8 ones + 5 ones = 13 ones!
Can we write "13" in the ones column?
NO! Only single digits (0 to 9) can live in the ones column!

The 10-for-1 Trade

Remember the golden rule of base-ten blocks: whenever you have 10 units, you must trade them for 1 rod! In 13 units, you have: 10 units (which becomes 1 NEW TEN) + 3 leftover units. You send the new ten over to live with the other tens, and keep the 3 ones in the ones column!

The Trading Transformation:
  13 Loose Units:   [#][#][#][#][#][#][#][#][#][#]   +   [#][#][#]
                    \__________________________/
                                 |
                            Trade for 1 Rod!
                                 v
                     [ 1 NEW TEN ROD ]               +   [ 3 ONES ]

The Vertical Regrouping Method

Here is how you write this trade using vertical addition notation:

Step-by-Step Vertical Regrouping for 28 + 15:

    [1]  <- This little "1" is the NEW TEN you composed!
     2  8
 +   1  5
 --------
     4  3

Step 1: Add the Ones: 8 + 5 = 13.
        13 is 1 ten and 3 ones.
        Write the 3 in the ones column below the line.
        Carry the 1 ten up to the top of the tens column!

Step 2: Add ALL the Tens (including the new ten!):
        1 + 2 + 1 = 4 tens!
        Write 4 in the tens column below the line.

Result: 43!

The Little 1 on Top

That little "1" perched at the top of the tens column is not just a scribble; it is the trophy of your trade! It represents the brand-new bundle of ten you created from the overflowing ones. Never forget to add it to the other tens!

Modeling with Base-Ten Blocks

Let us build 37 + 25 with physical rods and units.

Building 37 + 25:
37: 3 Rods and 7 Units
25: 2 Rods and 5 Units
------------------------------------
Gather the Units: 7 + 5 = 12 Units.
TRADE: Trade 10 units for 1 Rod!
Now you have: 2 units left, and 1 NEW Rod.
Gather all Rods: 3 original + 2 original + 1 NEW = 6 Rods!
Final Value: 6 Rods and 2 Units = 62!

Practice Exercises

Exercise 1: When adding the ones in 36 + 27, you get 6 + 7 = 13. Explain what you do with the 13 ones using the trading rule.

Exercise 2: Solve these vertical addition problems with regrouping: 48 29 56 35 + 25 + 34 + 17 + 28 ---- ---- ---- ----

Exercise 3: Solve 47 + 16. Show Step 1 (adding ones and trading) and Step 2 (adding tens).

Exercise 4: A student solved 28 + 14 and got 32. What mistake did the student make? What is the correct answer?

Solutions and Explanations

Solution to Exercise 1: You decompose 13 into 1 ten and 3 ones. You trade 10 of the units for 1 new rod of ten and place it in the tens column, leaving the 3 ones in the ones column.

Solution to Exercise 2: 48 + 25 = 73 (8+5=13; write 3, carry 1; 1+4+2=7 tens) 29 + 34 = 63 (9+4=13; write 3, carry 1; 1+2+3=6 tens) 56 + 17 = 73 (6+7=13; write 3, carry 1; 1+5+1=7 tens) 35 + 28 = 63 (5+8=13; write 3, carry 1; 1+3+2=6 tens)

Solution to Exercise 3: Step 1: Add ones: 7 + 6 = 13 ones. Regroup 13 into 1 ten and 3 ones. Write 3 in the ones place, and carry 1 ten to the tens place. Step 2: Add tens: 1 (carried) + 4 + 1 = 6 tens. The total sum is 63.

Solution to Exercise 4: The student added 8 + 4 = 12 and wrote down 2 in the ones place, but FORGOT to add the carried 1 ten to the tens column (adding only 2 + 1 = 3 instead of 1 + 2 + 1 = 4). The correct answer is 42.