Two-Digit Addition Without Regrouping

We start two-digit addition with the easy case: when the ones-place digits don't sum past 9.

What we're learning

That two-digit addition follows directly from place-value
The vertical "stacked" form (lining up tens and ones)
Why we add the ones first, then the tens

The easy case first

24 + 35 is the easy case. Why? Because 4 + 5 = 9 — no overflow. We can add the ones place straight up, then the tens place straight up, and we're done.

   2  4
 + 3  5
 ------
   5  9

Ones: 4 + 5 = 9. Tens: 2 + 3 = 5. Answer: 59.

The reason this works at all — and the reason your child needs to internalize it before moving to the harder case — is place value. The "2" in 24 means twenty. The "3" in 35 means thirty. Twenty plus thirty is fifty. The tens line just records that, in the tens column.

The order matters (and it's about to)

We add the ones first. In the easy case it doesn't seem to matter — you can add the tens first and get the same answer. But in the next lesson (regrouping), order becomes essential. Build the habit now.

Try it together

Get a pencil and notebook. Draw a vertical line down the middle of a column to make the place-value structure visible:

| Tens | Ones |
|  2   |  4   |
|  3   |  5   |
| ---  | ---  |
|  5   |  9   |

Try these together. Each one has no carrying:

13 + 25
41 + 38
60 + 27
52 + 14
70 + 20

The last one is interesting: 70 + 20 = 90. The ones are both 0. Show your child that nothing weird happens — 0 + 0 = 0, 7 + 2 = 9, answer 90.

Connecting back to expanded form

Watch this:

  24       (20 + 4)
+ 35       (30 + 5)
  --       --------
  59       (50 + 9) = 59

The vertical algorithm is doing what the expanded form makes obvious: add the tens (20 + 30 = 50), add the ones (4 + 5 = 9), put them back together. If your child has expanded form solid (last lesson of Place Value), this should feel like the same idea in a more compact notation.

Watch for

Adding columns the wrong way around. Some children, especially if they've been told to "start at the left" in reading, will try to add the tens first. Gently correct: "We always add the ones first. You'll see why next lesson."
Treating each column as separate. A child might write 24 + 35 as 5 9 (ones answer next to tens answer) without understanding that 5 means fifty. Use the place-value column drawing whenever this confusion shows up.
Skipping the column lineup. 24 + 5 (a 2-digit + a 1-digit) trips kids up because the lineup is not symmetric. Show that the 5 lines up under the 4 (ones place), not the 2 (tens place).

Where this is going

Next lesson: regrouping (also called "carrying"). What happens when the ones add up to 10 or more. That's where two-digit addition gets interesting — and where everything we've built about place value pays off.