6.3 Place Value & Base-Ten Operations

The Base-Ten System

Our number system is base ten (decimal): every 10 units of one place bundle into 1 of the next. A Georgia 2nd grader uses base-ten blocks (units, rods of 10, flats of 100, cubes of 1,000) to make this bundling physical. The GACE expects you to know both the positions and the 10:1 bundling ratio that defines them.

Place-Value Positions

Each period (ones, thousands, millions, billions) contains three positions: ones, tens, hundreds of that period. Commas in 4,308,217 separate the periods and make the number readable: 4 million, 308 thousand, 217.

Expanded Form and Regrouping

Expanded form shows the value of each digit:

4,308,217 = 4,000,000 + 300,000 + 8,000 + 200 + 10 + 7

A useful variant is expanded form with multiplication, sometimes called base-ten decomposition:

4,308,217 = 4 × 1,000,000 + 3 × 100,000 + 0 × 10,000 + 8 × 1,000 + 2 × 100 + 1 × 10 + 7 × 1

Regrouping (trading) is the inverse of bundling: 10 ones trade for 1 ten; 10 tens trade for 1 hundred; 10 hundreds trade for 1 thousand. Regrouping is the conceptual engine behind every standard algorithm for addition, subtraction, multiplication, and division.

Comparing and Rounding Multi-Digit Numbers

Comparing uses place value from left to right: 4,308,217 vs 4,308,099 — the millions, hundred-thousands, and thousands match; the hundreds digit (2 vs 0) decides. So 4,308,217 > 4,308,099.

Rounding uses the digit one place to the right of the target:

• Round 4,308,217 to the nearest thousand: look at the hundreds digit (2). Less than 5, so round down → 4,308,000.
• Round 4,308,217 to the nearest hundred-thousand: look at the ten-thousands digit (0). Less than 5, round down → 4,300,000.

The "5 or more, round up; 4 or less, round down" rule works because 5 is the halfway point in base ten.

Multi-Digit Operations

Addition and Subtraction with Regrouping

Standard algorithm for 358 + 274: add ones (8 + 4 = 12; write 2, regroup 1 ten), add tens (5 + 7 + 1 = 13; write 3, regroup 1 hundred), add hundreds (3 + 2 + 1 = 6). Answer: 632.

Partial sums (an intermediate strategy) records the sums by place value before combining:

358 + 274 300 + 200 = 500 50 + 70 = 120 8 + 4 = 12 500 + 120 + 12 = 632

For subtraction, the standard algorithm trades down when a digit is too small: 524 − 187 → ones: 4 < 7, trade 1 ten for 10 ones (14 − 7 = 7); tens: now 1 ten, 1 < 8, trade 1 hundred (11 − 8 = 3); hundreds: 4 − 1 = 3. Answer: 337.

Multi-Digit Multiplication

The area model connects the distributive property to multiplication. For 24 × 13:

200 + 40 + 60 + 12 = 312

Partial products records the same four products in a column:

24 × 3 = 72 24 × 10 = 240 72 + 240 = 312

Standard algorithm: 24 × 13 → multiply by 3 ones (72), multiply by 1 ten (240, shifted left), add (312). The shift left encodes "1 ten × 24 = 24 tens."

Long Division: Partial Quotients and Standard Algorithm

For 936 ÷ 4:

Partial quotients (more accessible) peels off friendly chunks:

4 × 200 = 800 → 936 − 800 = 136 4 × 30 = 120 → 136 − 120 = 16 4 × 4 = 16 → 16 − 16 = 0 Quotient: 200 + 30 + 4 = 234

Standard algorithm records this in the more compact divide-multiply-subtract-bring-down form, but the underlying reasoning is identical: find how many groups of 4 fit in 936. The partial-quotients method exposes that reasoning; the standard algorithm compresses it.

Decimals: Extending Place Value to the Right

Decimals are not a new system — they continue the 10:1 ratio to the right of the ones place, using a decimal point as the anchor.

Decimal Place-Value Positions

So 3.407 means 3 ones, 4 tenths, 0 hundredths, 7 thousandths. Expanded: 3 + 0.4 + 0.007 = 3.407.

A common 4th-grade misconception: thinking 0.407 is "larger than 0.5" because 407 > 5. The fix is to read decimals as fractions of a whole: 407 thousandths is less than 500 thousandths (0.5).

Decimal Operations

Addition and subtraction line up the decimal point so that like places combine with like places:

3.4 + 2.17 → 3.40 + 2.17 = 5.57

The decimal point in the answer lines up with the decimal points above.

Multiplication does not line up the points; it counts total decimal places. For 0.3 × 0.4:

• 3 × 4 = 12
• One decimal place in each factor → two decimal places in the product
• 0.3 × 0.4 = 0.12

Conceptually, 3 tenths of 4 tenths is 12 hundredths — the area of a 0.3-by-0.4 rectangle.

Division by a whole number keeps the decimal point aligned: 4.8 ÷ 4 = 1.2 (share 4 ones among 4 = 1; share 8 tenths among 4 = 2 tenths). When the divisor is a decimal, multiply both numbers by a power of 10 to make the divisor whole: 1.2 ÷ 0.3 = 12 ÷ 3 = 4. This preserves the quotient because multiplying both numbers by the same power of 10 does not change their ratio.