# Lesson 14

Using Diagrams to Represent Addition and Subtraction

### Lesson Narrative

Prior to grade 6, students have added and subtracted decimals to the hundredths using a variety of methods, all of which focus on understanding place value. This lesson reinforces their understanding of place-value relationships in preparation for computing sums and differences of any decimals algorithmically.

In this lesson, students use two methods—base-ten diagrams and vertical calculations—to find the sum and differences of decimals. Central to both methods is an understanding about the meaning of each digit in the numbers and how the different digits are related. Students recall that we only add the values of two digits 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 different 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 is made explicit both in the diagrams and in vertical calculations.

“Unbundling,” which students have previously used to subtract whole numbers, is also 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).

### Learning Goals

Teacher Facing

• Compare and contrast (orally and in writing) vertical calculations and base-ten diagrams that represent adding and subtracting decimals.
• Comprehend the term “unbundle” means to decompose a larger base-ten unit into 10 units of lower place value (e.g., 1 tenth as 10 hundredths).
• Explain (in words and through other representations) that adding and subtracting decimals requires combining digits that represent like base-ten units.
• Interpret and create diagrams that represent 10 like base-ten units being composed into 1 unit of higher place value, e.g., 10 tenths as 1 one, and comprehend the word “bundle” to refer to this concept.

### Student Facing

Let’s represent addition and subtraction of decimals.

### Required Preparation

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/NJPVks38.

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: Finding Sums in Different Ways, Subtracting Decimals of Different Lengths, and Why or Why Not?

### Student Facing

• I can use diagrams and vertical calculations to represent and reason about addition and subtraction of decimals.
• I can use place value to explain addition and subtraction of decimals.
• I know how to solve subtraction problems with decimals that require “unbundling” or “decomposing.”

Building On