Expanded Form

We make the place-value structure of a number visible by writing it as a sum of its parts.

What we're learning

The phrase "expanded form" — what it means and why
How to write any two-digit number as tens + ones
The reverse trick: given 40 + 6, what number is that?

What expanded form looks like

37 = 30 + 7
58 = 50 + 8
20 = 20 + 0
74 = 70 + 4

That's it. We write the tens place as a multiple of ten, then add the ones place.

The point isn't memorization. The point is to make the place-value structure of every number physically obvious on the page. Once your child can read 58 and instinctively think 50 + 8, multi-digit addition becomes much less mysterious.

Try it together

You'll need a notebook and a pencil.

Round 1: from number to expanded.

Write 46. Ask: "Can we break this into tens and ones?" Together, write: 46 = 40 + 6.

Do this 10 times with different two-digit numbers. Mix in 30 (= 30 + 0) and 19 (= 10 + 9) so they don't think only "round" numbers work.

Round 2: from expanded back to a number.

You write 60 + 5. They write 65. Then they write the next problem for you.

Round 3: with three parts. (Saved for after both directions feel easy.)

Add a hundreds place: 123 = 100 + 20 + 3. This makes the leap from two-digit to three-digit feel like the SAME idea, just one more column.

Why this isn't busywork

Expanded form is the missing rung on the ladder between "memorizing how to add columns" and "actually understanding why the columns work."

Consider 25 + 13:

Without expanded form, the child has memorized: "line them up, add the ones, add the tens." Brittle.
With expanded form, the child can write: (20 + 5) + (10 + 3) = (20 + 10) + (5 + 3) = 30 + 8 = 38. Understandable, derivable from first principles.

Both reach 38. But the second child can extend the same reasoning to 125 + 47 without being taught a new procedure. The first has to be re-taught.

Watch for

Writing 46 = 4 + 6 instead of 40 + 6. The leftmost digit's value is forty, not four. This is the place-value mistake we're trying to prevent. Catch it gently every time.
Skipping the zero in round numbers. 30 = 30 + 0. The zero feels weird to write. Insist on it for a while; the symmetry pays off when they hit numbers like 108 = 100 + 0 + 8.

Where this is going

The third and final lesson in this module: comparing two-digit numbers — using place value (look at the tens first!) to decide which of two numbers is bigger. After that, the door opens to two-digit addition and subtraction with regrouping.