The purpose of this activity is to examine subtraction cases in which non-zero digits are subtracted from zero digits. In some cases, students could simply look at the digit to the left of a 0 and decompose 1 unit of that number. But in other cases, the digit to the left is another 0 (or more than one 0), which means looking further to the left until reaching a non-zero digit. Students learn to decompose that unit first, and then move to the right, decomposing units of smaller place values until reaching the original digits being subtracted. The problems are sequenced from fewer zeros to more zeros to allow students to see how to successively decompose units.

Recording all of the decompositions can be challenging. For the last problem, two sample responses are given to show two different ways of recording the decompositions. The important point to understand is that because there are no tens, hundreds, or thousands to decompose, a ten-thousand must be decomposed to make 10 thousands. Then one of the thousands is decomposed to make 10 hundreds, and so on, until reaching the ones place. Those successive decompositions can be lined up horizontally, but this can make it hard to see what happened first. A second way shows more clearly the order in which the decompositions occur, but it may be challenging to see which place the successive decomposed units are in.

To add movement to this activity, the second problem could be done as a Gallery Walk where each group completes one problem and then walks around the room to look for similarities and differences in others’ posters.

• Ask students to share responses and to demonstrate consecutive decomposing when multiple zeros are involved.
• If needed, use expanded form to represent the decompositions students are explaining.

In this activity, students solve contextual problems that involve subtracting numbers with non-zero digits from numbers with one or more zero digits. Students may choose other ways to find the difference (for example, adding up and using a number line to keep track), but are asked to use the standard algorithm at least once.

In the Launch, students subtract their age from the current year. This provides an opportunity for students to notice the relationship between this difference and their current age (MP7).

• Ask students to share responses.
• “Which years did you find easiest to subtract? Which were more difficult? Why?” (Subtracting a year without decomposing any units was easier than subtracting a year that required decomposing.)
• Prompt students to check and compare ages found using the standard algorithm and those found using other ways of reasoning.