Lesson 3

As with addition, prior to grade 6 students have used various ways to subtract decimals to hundredths. Base-ten diagrams and vertical calculations are likewise used for subtracting decimals. “Unbundling,” which students have previously used to subtract whole numbers, is a key idea here. They recall that a base-ten unit can be expressed as another unit that is \(\frac{1}{10}\) its size. For example, 1 tenth can be “unbundled” into 10 hundredths or into 100 thousandths. Students use this idea to subtract a larger digit from a smaller digit when both digits are in the same base-ten place, e.g., \(0.012 - 0.007\). Rather than thinking of subtracting 7 thousandths from 1 hundredth and 2 thousandths, we can view the 1 hundredth as 10 thousandths and subtract 7 thousandths from 12 thousandths. 

Unbundling also suggests that we can write a decimal in several equivalent ways. Because 0.4 can be viewed as 4 tenths, 40 hundredths, 400 thousandths, or 4,000 ten-thousandths, it can also be written as 0.40, 0.400, 0.4000, and so on; the additional zeros at the end of the decimal do not change its value. They use this idea to subtract a number with more decimal places from one with fewer decimal places (e.g., \(2.5 - 1.028\)). These calculations depend on making use of the structure of base-ten numbers (MP7).

The second activity is optional; it gives students additional opportunities to practice summing decimals.

Students draw base-ten diagrams in this lesson. If drawing them is a challenge, consider giving students access to:

• Commercially produced base-ten blocks, if available.
• Paper copies of squares and rectangles (to represent base-ten units), cut outs from copies of the blackline master of the second lesson in the unit.
• Digital applet of base-ten representations https://ggbm.at/zqxRkhMh.

Some students might find it helpful to use graph paper to help them align the digits for vertical calculations. Consider having graph paper accessible for these activities: Representing Decimal Subtraction and Enough to Subtract?