This activity continues to reinforce students’ understanding of place value in base-ten numbers. Namely, that the value of a digit is 10 times the value of the same digit immediately to its right (and the value of the same digit immediately to its left).

Students see—in both the diagrams and the numbers being represented—that 10 hundredths has the same value as 1 tenth, that 10 thousandths has the same value as 1 hundredth, and so on. They use this understanding to create equivalent diagrams that represent decimals to thousandths.

In the digital version of the activity, students use an applet to create base-ten diagrams that represent numbers to two decimal places. The applet allows students to generate blocks that represent ones, tenths, and hundredths, but doesn’t include blocks for thousandths, which would not be large enough to manipulate. It also allows students to compose 10 hundredths into 1 tenth and 10 tenths into 1. The digital version may reduce barriers for students who need support with fine-motor skills.

The goal of this discussion is to highlight that:

• A base-ten unit can be decomposed into 10 units in the place immediately to its right.
• A group of 10 base-ten units can be composed into 1 larger unit in the place immediately to its left.

Invite students to share their diagrams or display the solutions for all to see. Discuss questions such as:

• “Why might it be helpful to use as few base-ten blocks as possible when representing a number?” (Fewer pieces mean less counting and a smaller likelihood of making a counting mistake.)
• “When is it possible to use fewer shapes or blocks to represent a number?” (When there are at least 10 of a unit, the 10 units can be composed into 1 larger unit.)
• “What would a diagram that represents 10 look like?” (An extra-large rectangle, composed of 10 big squares.) “What about 100?” (A giant square made from 10 of the extra-large rectangles representing 10.)
• “Why might it be inconvenient to use base-ten diagrams to represent larger multi-digit numbers or numbers with more decimal places?” (It’d mean drawing a lot of squares and rectangles. When there are more units, the diagrams become more complex.)

In this activity, students use diagrams and the standard algorithm to find a sum that requires regrouping of base-ten units. Students see in both representations how 10 smaller base-ten units can be composed into a larger one. Along the way, students practice looking for and making use of structure (MP7).

To illustrate the idea of composing and decomposing more concretely, use of physical base-ten blocks or paper cutouts of base-ten representations (from the blackline master) is recommended.

In the digital version of the activity, students use an applet to create base-ten diagrams that represent sums of numbers to two decimal places. The applet allows students to generate blocks that represent ones, tenths, and hundredths, but doesn’t include blocks for thousandths, which would not be large enough to manipulate. It also allows students to compose 10 hundredths into 1 tenth, and 10 tenths into 1. The digital version may reduce barriers for students who need support with fine-motor skills.

Focus the discussion on the connections across the two representations of addition of decimals. Discuss questions such as:

• “When adding 0.38 and 0.69, in which place(s) was it possible to compose a unit from 10 of a smaller unit?” (In the hundredth and tenth places) “Why?” (There were a total of 17 hundredths, and 10 hundredths can be composed to make 1 tenth. This 1 tenth is added to the 3 tenths and 6 tenths, which makes 10 tenths. Ten tenths can be composed into 1 one.)
• “Where can we see the process of composing a larger unit in the vertical calculations?” (The 1 in the tenths place, above the 3 in 0.38, was composed of 10 of the 17 hundredths from adding 8 hundredths and 9 hundredths. The 1 in the ones place was composed of 10 of the 10 tenths from adding 6 tenths and 3 tenths in the numbers being added, plus the 1 tenth composed from 10 hundredths.)
• “Which method of calculating is more efficient?” (It depends on the complexity and size of the numbers. The drawings become hard when there are lots of digits or when the digits are large. The algorithm works well in all cases, but it is more abstract and requires that all bundling be recorded in the right places.)

Highlight the idea of choosing appropriate tools to solve a problem, which is an important part of doing mathematics. Point out that physical blocks or base-ten diagrams can effectively help us understand what is happening when we add base-ten numbers, before moving on to a more generalized method.

In this activity, students use base-ten diagrams and vertical calculations to perform subtraction. As with addition of decimals, students need to pay close attention to place value when calculating differences. They identify the need to pair the digits of like base-ten units when subtracting decimals and why it is helpful to line up the decimal points. The subtractions in this activity do not require regrouping or decomposing a larger unit into 10 of a smaller unit.

The goal of the discussion is to make sure students understand that when we perform subtraction without diagrams, it is essential to pay close attention to place value in the numbers.

Select a few students to share their responses and reasoning for the last two questions. Display their diagrams and calculations for all to see. Discuss questions such as:

• “How are addition and subtraction of decimal numbers alike?” (It is important to attend to place value and to add or subtract numbers that represent the same base-ten units.)
• “Why is it helpful to line up the decimal points when calculating differences of decimals?” (Aligning the points helps us align digits with the same place value.)
• “Which is more efficient for subtracting decimals, using base-ten blocks or using vertical calculations?” (For some numbers, such as , both methods are efficient. If the numbers contain more decimal places or larger digits, the diagrams would take a lot of time to draw.)