Numerical Reasoning, Patterns, and Algebraic Thinking · 3 min read
Base-Ten Place Value and Decimals
The base-ten system uses ten digits, 0 through 9, and assigns each digit a value determined by its position. Each place is worth ten times the place to its right and one-tenth of the place to its left. This single rule governs whole numbers and decimals alike: the tenths place is one-tenth of the ones place, and the hundredths place is one-tenth of the tenths place. The digit 0 acts as a placeholder, so 507 and 57 are different numbers.
The face value of a digit is the digit itself; its place value is the digit times the value of its position. In 63,482 the digit 4 has face value 4 and place value 400. Comparing the same digit in two positions shows the multiplicative structure: the 5 in 3,450 is worth 50, and the 5 in 3,045 is worth 5, so the first is ten times the second.
Models and Representations
Base-ten blocks (units, rods, flats, cubes), place-value charts, bundled sticks, ten-frames, hundred charts, and decimal grids all display the same structure. Grouping models such as bundled sticks are made by the child and are concrete; pre-grouped models such as base-ten blocks are faster to use but require the child to accept that one rod equals ten units.
| Form | Representation |
|---|---|
| Standard form | 5,208.36 |
| Word form | five thousand, two hundred eight and thirty-six hundredths |
| Expanded form | 5,000 + 200 + 8 + 0.3 + 0.06 |
| Expanded notation with multiplication | (5 × 1,000) + (2 × 100) + (0 × 10) + (8 × 1) + (3 × 0.1) + (6 × 0.01) |
| Exponent form of the powers | (5 × 10³) + (2 × 10²) + (8 × 10⁰) + (3 × 10⁻¹) + (6 × 10⁻²) |
In word form, the word and marks the decimal point and should not be used within the whole-number part; 208 is read "two hundred eight," not "two hundred and eight."
Regrouping and Nonstandard Partitions
Numbers can be decomposed in nonstandard ways that keep the same value: 352 is 3 hundreds 5 tens 2 ones, or 2 hundreds 15 tens 2 ones, or 35 tens 2 ones. Recognizing these partitions is the basis of regrouping in subtraction (trading 1 ten for 10 ones) and in addition (trading 10 ones for 1 ten).
Multiplying or dividing by powers of ten moves each digit to a new place: 3.7 × 100 = 370 because each digit moves two places to the left, and 52 ÷ 1,000 = 0.052 because each digit moves three places to the right.
Tenths and hundredths can be modeled with a 10 × 10 grid as the whole: one column is 0.1 and one small square is 0.01. Shading 0.35 colors 3 full columns and 5 extra squares, which displays both the decimal and its equivalent fraction 35/100.
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