Place Value

Place value is the idea that our number system uses position to encode value. The same digit means different things depending on where it sits — the 4 in 4, 40, and 400 represents four ones, four tens, and four hundreds. Kids learn to read multi-digit numbers, decompose them into ones, tens, hundreds, and so on, compare numbers, and eventually extend the idea to decimals.

It's taught from kindergarten (two-digit numbers, tens and ones) through third grade (thousands, then place value into the decimals in fourth and fifth). It is the single most important concept of early elementary math. Every standard algorithm — adding with carrying, subtracting with borrowing, multi-digit multiplication, long division, decimals — assumes a kid understands place value cold. Most computation struggles in second through fifth grade trace back to shaky place value, not the procedure itself.

The trap with place value is that it looks easy. A kid can recite ones, tens, hundreds and seem to have it, then completely fall apart when asked what number 4 tens and 13 ones makes, or why you carry the 1. Verbal recitation is not understanding.

What actually builds place value:

• Base-ten blocks, popsicle sticks bundled in tens, or any physical manipulative where ten ones literally become one ten. Kids need to see the swap with their hands before they trust it on paper.
• Lots of decomposing — write 47 as 40 + 7, then as 30 + 17, then as 20 + 27. Flexibility with regrouping is what carries them through borrowing and into mental math.
• Math Mammoth, Singapore Math, and Beast Academy all handle place value well; most app-only programs rush it.

Gut-check: ask your kid to show you 42 in two different ways using blocks or drawings. If they can do 4 tens and 2 ones, then also 3 tens and 12 ones, they get it. If they can only do the first one, they're memorizing.