In this lesson, students use base-ten diagrams and vertical calculations to add and subtract decimals to thousandths. Central to both methods is an understanding about the meaning of each digit in the numbers and how the different digits are related.

First, students recall that we add the values of two digits only if they represent the same base-ten units. They also recall that when the value of a base-ten unit is 10 or more, we can express it with a larger unit that is 10 times higher in value. For example, 10 tens can be expressed as 1 hundred, and 12 hundredths can be expressed as 1 tenth and 2 hundredths. This idea becomes more explicit to students as they make use of the structure in the diagrams and in vertical calculations (MP7).

Next, students recall that a base-ten unit can be decomposed into another unit that is its size. For example, 1 tenth can be decomposed into 10 hundredths. They use this idea to subtract a larger value from a smaller value that is in the same base-ten place. Take for instance. Decomposing the 1 hundredth in 0.012 as 10 thousandths makes it easier to subtract 7 thousandths.

In all activities in the lesson, students are prompted to create visual representations of base-ten units. This can be done by drawing base-ten diagrams, using physical base-ten blocks (if available), or using paper cutouts of base-ten representations from the blackline master. Consider providing access to these tools as alternatives to drawing.

Some students might find graph paper helpful for aligning the digits for vertical calculations. Consider making graph paper accessible for activities that involve addition and subtraction of decimals.