Connect tenths, hundredths and decimal notation

Learning goal: Rename tenths as hundredths, add tenths and hundredths, and use decimal notation to describe and locate fractional amounts. A whole can be partitioned into ten equal tenths or one hundred equal hundredths. Every tenth contains ten hundredths. The decimal point marks the boundary between whole units and fractional place values.

The square grid in the illustration contains ten rows of ten equal cells. The entire square is one whole. Each cell is one hundredth, and each full row is one tenth. Three shaded full rows represent 3/10 = 30/100. The amount does not change when every tenth is subdivided into ten hundredths. Count thirty selected cells, not just three selected rows, when using hundredths.

Suppose a length is 3/10 meter and another is 4/100 meter. To combine them, use the same piece size. Rename three tenths as thirty hundredths, then add 30/100 + 4/100 = 34/100 meter. Adding numerators before renaming would combine counts of unequal units. Adding denominators to get 7/110 would not describe the pieces in either measurement.

Decimal notation writes tenths immediately to the right of the decimal point and hundredths one place farther right. Thus 3/10 is 0.3, 4/100 is 0.04 and 34/100 is 0.34. Read 0.34 as thirty-four hundredths. It contains three tenths and four additional hundredths. The zero in 0.04 shows that there are no full tenths; omitting it would change the amount to four tenths.

A trailing zero can preserve value. The decimal 0.3 means three tenths, and 0.30 means thirty hundredths. These are equivalent amounts, so 0.3 = 0.30. The extra written digit does not automatically make the number larger. In contrast, 0.03 means three hundredths, which is only one tenth of three tenths.

A decimal can describe a measured length. A mark at 0.62 meter is sixty-two hundredths of a meter from the zero endpoint. Since one meter contains one hundred centimeters, that is sixty-two centimeters. On a number line from zero to one meter, first find six tenths, or 0.60, then move two hundredth intervals farther. The point lies between 0.60 and 0.70 and nearer 0.60.

Each tenth interval on that line can be split into ten equal hundredth intervals. The screen drawing is a proportional model, not a physical ruler whose pixels equal centimeters. For a real measuring task, use an actual meter stick or tape. Align the object's start with zero, read the endpoint and state the unit in both fractional and decimal forms.

This lesson combines denominators ten and one hundred by their place-value relationship. For example, 7/10 + 8/100 = 70/100 + 8/100 = 78/100 = 0.78. The reason is the common hundredth unit. A memorized decimal-point procedure without that unit explanation does not show why the answer makes sense.

Worked conversion: Write 0.62 as a fraction: 62/100. Decompose it as 60/100 + 2/100 = 6/10 + 2/100. On a meter line, it is two hundredths past six tenths. Its location must be beyond one half but before seven tenths.

Worked sum: Join 6/10 meter and 17/100 meter. Rename 6/10 as 60/100, then add 60/100 + 17/100 = 77/100 meter, or 0.77 meter. The total is larger than six tenths but less than one whole meter.

Guided practice: Shade full rows and individual cells of a hundred grid. Record each amount as tenths plus hundredths, a fraction with denominator one hundred and a decimal. Place the matching point on a zero-to-one line with equal subdivisions.

Independent evidence: Measure an object shorter than one meter to the nearest centimeter with a real tool and record its length in hundredths of a meter and decimal meters. Then locate 0.62 on a drawn number line and solve a new tenths-plus-hundredths sum. Record tool alignment and explanation separately from numerical accuracy; an imagined measurement is not evidence of using the tool.