Compare decimal amounts using place value and models

Learning goal: Compare decimals through hundredths, explain their order using place value or a model and record the result with <, > or =. Decimal notation names fractions using the same whole-unit reference. In an area model the whole is the entire grid; in a meter model it is one meter; on a number line it is the distance from zero to one.

Compare 0.7 and 0.65. Seven tenths equal seventy hundredths, so rename 0.7 as 0.70. Both values now count hundredths: seventy is greater than sixty-five, so 0.7 > 0.65. On matching hundred grids, the first amount covers five more cells. On a shared number line, 0.70 lies five hundredth intervals to the right of 0.65.

The number of digits after the decimal point does not determine the size. The numeral 0.65 has more decimal digits than 0.7 but names a smaller number. Read the place values: six tenths and five hundredths do not reach seven tenths. Applying a whole-number comparison to 65 and 7 ignores the units represented by those digits.

Compare 0.46 and 0.49. Both have four tenths. The hundredths differ: six hundredths are less than nine hundredths, so 0.46 < 0.49. The difference is three hundredths. When whole-number parts and tenths match, the hundredths place can decide the order. Compare the greatest place first, because smaller places cannot compensate for a full difference in a greater place within these place-value positions.

Zeros require attention. The decimals 0.08 and 0.8 mean eight hundredths and eight tenths. Rename eight tenths as eighty hundredths: 0.08 < 0.80. The zero immediately after the decimal point in 0.08 holds the tenths place. Removing it changes the value. By contrast, the trailing zero in 0.80 does not change the amount because eighty hundredths equal eight tenths.

Equality is also a possible conclusion. Three tenths and thirty hundredths cover the same part of a whole, so 0.3 = 0.30. If you draw their points on the same number line, both belong at the same location. Different notation can name one value, just as one half and two fourths can.

Compare whole-number parts when the amounts exceed one. One and two hundredths, written 1.02, is greater than 0.98 because it passes one whole while ninety-eight hundredths does not. If both amounts have the same whole-number part, compare tenths and then hundredths. For example, 1.09 < 1.2 because 1.2 = 1.20 and nine hundredths is less than twenty hundredths beyond the common whole.

A contextual comparison must keep the unit and whole clear. It is valid to compare 0.7 meter and 0.65 meter directly. But a drawing showing 0.7 of a small garden and another showing 0.65 of a larger garden cannot establish which selected area has more square meters. The fraction values are ordered, while the physical areas also depend on each garden's whole area.

Worked order: Arrange 0.5, 0.05 and 0.45 from least to greatest. Rename 0.5 as 0.50. The hundredth counts are 50, 5 and 45, so 0.05 < 0.45 < 0.5. Explain that five hundredths is much smaller than five tenths, even though both numerals contain a five.

Worked justification: A jump measures 0.68 meter and another 0.6 meter. Six tenths equal sixty hundredths. Sixty-eight hundredths is eight hundredths greater, so 0.68 > 0.6. A shared meter number line shows the first endpoint farther from zero.

Guided practice: Use matching hundred grids to compare pairs with different tenths, equal tenths and equal values. Say each decimal in words before choosing a symbol. Include a deliberate incorrect claim based on digit count and ask the learner to refute it with the selected regions.

Independent evidence: Order three unfamiliar decimals, justify one inequality with a number line and explain one equality with equivalent fractions. Then compare two measured lengths in the same unit. Require an explanation of the zero's role in 0.08 versus 0.8 and of why differently sized whole objects need more information.