Explain decimal places and powers of ten

Learning goal: Explain the ten-to-one relationship between adjacent places and use whole-number exponents for powers of ten. Track the value represented by a digit, not just its appearance on the page. A decimal point locates the boundary between ones and fractional places.

In 444.4, the four in the hundreds place represents 400, the four in the tens place represents 40, the four in the ones place represents 4 and the four in the tenths place represents 0.4. Each value is ten times the value immediately to its right. It is also one tenth of the value immediately to its left. The digit is the same, but its place determines the amount represented.

A whole divided into ten equal parts gives tenths. Dividing each tenth into ten equal parts gives hundredths, so the whole contains one hundred hundredths. Dividing each hundredth into ten equal parts gives thousandths. The unit named by the place gets smaller toward the right. Ten thousandths compose one hundredth; ten hundredths compose one tenth; ten tenths compose one whole.

A power of ten records repeated multiplication by ten. 10¹ = 10, 10² = 10 × 10 = 100 and 10³ = 10 × 10 × 10 = 1,000. The small raised whole number is an exponent telling how many factors of ten occur in these examples. It does not tell you to multiply ten by that number: 10³ is not 30.

Multiplying 4.37 by 10 makes every represented amount ten times as large: four ones become four tens, three tenths become three ones and seven hundredths become seven tenths. The result is 43.7. Multiplying by 100 makes the result 437; multiplying by 1,000 gives 4,370. A place-value chart explains the changing positions relative to the decimal point. In a fixed chart, the decimal point stays at the ones boundary while the digits occupy different places.

Dividing reverses that relationship. 43.7 ÷ 10 = 4.37 and 43.7 ÷ 100 = 0.437. Each represented amount becomes one tenth as large for each division by ten. A zero before the decimal point makes clear that a number such as 0.437 has no complete ones. Zeroes can also occupy empty internal places so that the other digits keep their intended values.

For whole numbers, multiplication by powers of ten often creates trailing zeroes. 56 × 100 = 5,600 because five tens and six ones become five thousands and six hundreds. There are no tens or ones left in the result's smaller places, so zeros hold those places. The exponent predicts the pattern: 56 × 10¹ = 560, 56 × 10² = 5,600 and 56 × 10³ = 56,000. For a whole number, each factor of ten shifts every digit one place left, so the product ends in as many added zeros as the exponent. For a decimal, the same shift shows up as the digits moving that many places relative to the decimal point: 4.37 × 10² = 437 and 437 ÷ 10² = 4.37. Blindly adding zeros fails for decimals: writing 4.3700 after multiplying 4.37 by 100 has not changed its value at all.

Use size as a check. Multiplying a positive number by 10 should make it greater; dividing by 10 should make it smaller. If 0.82 × 100 is written as 0.0082, the direction is wrong. Explain the change in units and estimate the size before committing to a written answer.

Worked expansion: 6.205 × 10 = 62.05. Six ones become six tens; two tenths become two ones; zero hundredths becomes zero tenths; five thousandths becomes five hundredths. The internal zero remains necessary to show that there are no tenths.

Worked exponent: 3.6 × 10² = 3.6 × 100 = 360. The exponent represents two factors of ten. Explain the two successive tenfold changes instead of replacing the exponent with a multiplication by two.

Guided practice: Use digit cards in a place-value chart to represent a number, then a tenfold larger and tenfold smaller number. Name the value of every nonzero digit after each change. Include a number with an internal zero.

Independent performance: Create three related equations using a decimal and powers 10¹, 10² and 10³. Explain one multiplication and one division with expanded values or a chart. Diagnose a mistaken “append zeros” answer. Assess the values and reasoning rather than accepting a rule about moving a dot without explanation.