Read, expand and compare decimals to thousandths

Learning goal: Read and write decimals through thousandths, express them in expanded form and compare them using place values. A longer decimal is not automatically greater. The units represented by the digits decide the comparison.

Consider 347.392. It contains three hundreds, four tens, seven ones, three tenths, nine hundredths and two thousandths. Read it as three hundred forty-seven and three hundred ninety-two thousandths. The word and marks the whole and fractional portions in this number name. The fractional part is 392/1,000, not 392 whole units.

Expanded form connects every digit to its value: 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1,000). You may also write 300 + 40 + 7 + 0.3 + 0.09 + 0.002. Both forms preserve the same total. Expanded form is useful for seeing a missing place that a spoken list of digits might hide.

For 5.047, the zero indicates that there are no tenths. The fractional value is forty-seven thousandths, or 0.04 + 0.007. Writing 5.47 would give four tenths and seven hundredths instead. A zero between digits can be essential. Writing 5.0470, however, would name the same value using an additional fractional place; this lesson works through thousandths.

Compare decimals from the greatest place toward the smallest. In 4.372 and 4.327, the ones and tenths match. The hundredths differ: seven hundredths is greater than two hundredths. Therefore 4.372 > 4.327, regardless of the thousandths. For two positive decimals in the same unit, lined up by place value, the first place where the digits differ settles the comparison.

Compare 0.8 and 0.75 by renaming 0.8 as 0.800 and 0.75 as 0.750. Eight tenths is greater than seven tenths, so 0.8 > 0.75. Having more written decimal digits does not make 0.75 larger. Similarly, 0.406 < 0.46 because the hundredths are zero and six after the matching four tenths. Trailing zeroes can help align places without changing values.

A number line makes the unit size visible. Between 0.37 and 0.38 there are ten equal thousandth intervals. Their endpoints are 0.370, 0.371, 0.372 and so on through 0.380. The number 0.376 lies six intervals beyond 0.370 and four intervals before 0.380. It is greater than 0.374 because it is farther to the right on this increasing line.

Always retain the measurement unit in a context. A ribbon measuring 0.406 meter is shorter than one measuring 0.460 meter when both use meters. Comparing 0.406 meter with 0.460 centimeter by decimal digits alone would compare unlike units. First establish that the quantities refer to the same unit before applying the decimal comparison.

Worked number name: Six and eight thousandths is 6.008. The two zeroes reserve the tenths and hundredths places. Writing 6.8 would mean six and eight tenths, a different number.

Worked comparison: 12.090 = 12.09 because nine hundredths equals ninety thousandths. Both exceed 12.009, which has only nine thousandths after the same twelve whole units. Explain with units before recording the symbols.

Guided practice: Match number names, decimals and expanded expressions on cards. Include internal zeroes and equivalent values with trailing zeroes. Place several thousandth values between consecutive hundredths on a labeled line.

Independent performance: Record three fictional measurements to thousandths with the same unit, write their expanded forms and arrange them from least to greatest. Defend two comparisons using the first differing place. Create an example that disproves “more digits means larger.” Check that the line's intervals are equal and that each number name matches the recorded value.