4.A.1 — Read and write multi-digit numbers (to 1,000,000).
Step 01 of 04
Big numbers are organized into periods of three digits, separated by commas. Each period has its own ones, tens, hundreds — and a name.
| Ones period | →...,XYZ — ones, tens, hundreds |
| Thousands period | →XYZ,... — thousand, ten thousand, hundred thousand |
| Millions period | →XYZ,XYZ,XYZ — million, ten million, hundred million |
Step 02 of 04
Worked example. Read 437,592.
Split at the comma: 437 thousand, then 592. Read each piece, then say the period name where the comma is.
437,592→four hundred thirty-seven THOUSAND, five hundred ninety-two
Step 03 of 04
Write from words. "Two hundred fifty-six thousand, eighteen."
Two hundred fifty-six → 256. Comma. Eighteen → 018 (must be three digits in the period — pad with a zero).
256,018
Step 04 of 04
Standard, word, and expanded form.
| Standard form | →4,275 |
| Word form | →four thousand, two hundred seventy-five |
| Expanded form | →4,000+200+70+5 |
Key insight
Three digits per period. Read each period normally, then say its name (thousand, million). When writing from words, every period gets exactly three digits — pad with leading zeros if needed.
4.A.2 — Compare multi-digit numbers.
Step 01 of 04
Step 1 — line up the place values. Make sure both numbers have the same number of digits (pad shorter ones with leading zeros mentally).
Example: compare 4,372 and 4,309. Both 4-digit numbers, no padding needed.
Step 02 of 04
Step 2 — compare digits left to right. Find the first place where they differ. Whichever number is bigger there is the bigger number overall.
| 4,372 | →thousands: 4 |
| 4,309 | →thousands: 4 — same |
| 4,372 | →hundreds: 3 |
| 4,309 | →hundreds: 3 — same |
| 4,372 | →tens: 7 |
| 4,309 | →tens: 0 — DIFFERENT, 7 > 0 |
So 4,372 > 4,309.
Step 03 of 04
Different number of digits → easy. The number with more digits is automatically bigger (as long as you don't write leading zeros).
12,450>9,999
Five digits beats four, no further comparison needed.
Step 04 of 04
Symbols.
| > | →greater than |
| < | →less than |
| = | →equal to |
Mnemonic: the open mouth always faces the bigger number. 5>3: open faces 5.
Key insight
Line up place values. Compare digit by digit, left to right. First different digit decides. If lengths differ, more digits wins.
4.A.3 — Round multi-digit numbers.
Step 01 of 04
Rounding replaces a number with a nearby "rounder" number. The question always specifies WHICH place value to round to.
Recipe: look at the digit RIGHT AFTER the rounding place. If it's 5 or greater, round UP. If it's less than 5, round DOWN.
Step 02 of 04
Worked example. Round 4,372 to the nearest hundred.
The hundreds digit is 3. The digit after it (tens) is 7. Since 7 ≥ 5, round UP: hundreds digit goes from 3 to 4.
4,372→4,400
Everything to the right of the rounding place becomes zero.
Step 03 of 04
Rounding down. Round 4,372 to the nearest thousand.
The thousands digit is 4. Next digit (hundreds) is 3. Since 3 < 5, round DOWN: thousands stays at 4.
4,372→4,000
Step 04 of 04
Rounding-up cascade. Sometimes rounding up forces multiple digits to change.
2,996→3,000(to nearest hundred)
The hundreds 9 + 1 = 10, which carries over: 2,9_96 → 3,0_00.
Key insight
Look at the digit IMMEDIATELY AFTER the rounding place. ≥5 round up, <5 round down. Replace everything to the right with zeros. Watch for cascading carries when rounding 9s up.
4.A.4 — Place-value relationships — the 10× pattern.
Step 01 of 04
In our number system, EACH place value is exactly 10 TIMES bigger than the place to its right.
| Tens | →10 times the ones |
| Hundreds | →10 times the tens, 100 times the ones |
| Thousands | →10 times the hundreds, 1000 times the ones |
| Each step left | →× 10 |
| Each step right | →÷ 10 |
Step 02 of 04
Worked example. The 7 in 7,000 is worth how many times the 7 in 700?
7,000/700=10
Ten times bigger. Equivalently, the 7 moved one place to the LEFT, which multiplies its value by 10.
Step 03 of 04
Same digit, different positions. In 5,555 the four 5s have four different values: 5, 50, 500, and 5000. Same digit, but each one is 10× the next as you move LEFT.
5,555=5,000+500+50+5
Step 04 of 04
Why this matters. Multiplying or dividing by 10, 100, or 1000 just SHIFTS digits — no calculation needed.
| 82×10 | →820 (shift left, fill with 0) |
| 82×100 | →8,200 |
| 820÷10 | →82 |
| 8200÷100 | →82 |
Key insight
Each place is 10× the one to its right. Same digit changes value dramatically by position. Multiplying by powers of 10 just shifts the digits left; dividing shifts them right.