This activity has two parts and serves two purposes. The first set of problems aims to help students see the limits of using base-ten diagrams to add and subtract numbers and to think about choosing more-efficient methods. The latter two questions prompt students to reason, in the context of situations, about addition and subtraction of numbers with more decimal places. In all problems, the numbers have enough decimal places that using base-ten representations (physical blocks, drawings, or digital representations) would be cumbersome, making numerical ways of reasoning appealing.

When students work on the first set of problems, monitor for those who use different strategies to subtract. Here are some likely strategies, from more elaborate to more streamlined:

• Using base-ten blocks or diagrams to represent the first number in each expression and removing pieces to represent subtraction.
• Writing each number in expanded form (decomposing each number by place value), subtracting the values in each place separately, and then adding them back at the end. (See sample reasoning for in Student Response.)
• Using vertical calculation or standard algorithm for subtraction.
• Reasoning about equivalence and place value, such as thinking of 0.02 and 0.0116 as 200 ten-thousandths and 116 ten-thousandths, and then subtracting the whole numbers, .

When working on the last two questions, students need to determine the appropriate operations for the given situations, perform the calculations, and then relate their answers back to the contexts. In doing so, they practice reasoning abstractly and quantitatively (MP2).

Focus the discussion on how students interpreted the last two questions, decided on what operations to perform, and found the sum or differences (including how they handled any regroupings). Select a student to share the response and reasoning for each question. Discuss questions such as:

• “When solving the word problems, how did you know whether to use addition or subtraction?”
• “What method did you use to find the sums or differences? Why did you choose that method?”

If some students chose base-ten diagrams for the calculations and if time permits, consider contrasting its efficiency with that of numerical calculations.

In this activity, students find differences of pairs of numbers with different numbers of decimal places, prompting them to look for and make use of structure (MP7). For instance, to find the difference of a whole number and a decimal, such as , they may:

• Rely on what they know about finding a difference of two whole numbers, such as .
• Find the difference of , and then add 0.001 to the difference.
• Subtract by the value of one place at a time, so first find , then , and finally .

The first two problems are presented as unknown-addend problems to activate what students know about the relationship between addition and subtraction to find differences (for instance, to think of in terms of ).

Select students who reason in different ways, and invite them to share later.

Ask previously selected students to share their responses and discuss their approaches: whether they work backward, write additional zeros, or use other strategies to find the unknown numbers.

If no students mentioned replacing the 0.7, 7, or 70 in the last three problems with 0.699, 6.9999, or 69.9999, discuss this idea. Ask students how their work to find would change if they were to replace 7 with 6.9999. Ask questions such as:

• “How might the number 6.9999 help us find the unknown number?” (The 9s make it easier to subtract, so we can find the difference between 6.9999 and 3.4567, and then add 0.0001 to the result because 6.9999 is 0.0001 less than 7.)
• “How would this method work for a problem such as ?” (We can replace 10 with 9.9999, determine the unknown number, and then add 0.0001 to that number).

This strategy is effective because it eliminates the “extra zeros” and the need to compose or decompose.