The purpose of this activity is for students to subtract in a way that makes sense to them. Students use an approach of their choice and share their approach with one another. This can serve as a formative assessment of how students think about finding the value of a difference when subtracting by place and a ten must be decomposed. Although students may use many ways to subtract, including those based on counting or compensation, the Synthesis focuses on connecting these approaches to those based on place value where a ten is decomposed.

Monitor and select students with the following approaches to share in the Synthesis:  

• Show 82 with connecting cubes, remove 9 cubes, and count all or count back from 82 to find the difference.
• Subtract 2 from 82 to get to a ten (80) and then subtract 7 from 80 by counting back (with or without connecting cubes or blocks).
• Show 82 with base-ten blocks, decompose a ten to get 12 ones, remove 9 ones from 12 ones, and count the remaining blocks.

The approaches are sequenced from more concrete to more abstract as a way to invite all students to make sense of subtraction approaches where a ten is explicitly decomposed. As an alternative, if the majority of the class uses the same approach (including one not listed above), consider sequencing the approaches from more common to less common (or less common to more common if the majority of students show decomposing with base-ten blocks). Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “How are these ways for finding the value of  the same? How are they different?” (They all started by showing 82. They all took away 9. They used different ways to represent 82. They counted differently. Some counted back, some removed some blocks and counted all. The base-ten blocks showed decomposing a ten.)
• Connect students’ approaches to the learning goal by asking:
  • “Why did _____ need to change 82 to 7 tens and 12 ones to subtract ones from ones?” (There are too many ones to take away, so she traded a ten for 10 ones to have enough ones to take away 9.)

The purpose of this activity is for students to subtract a one-digit number from a two-digit number with decomposing. In the previous activity, students shared many ways to subtract, including using connecting cubes or base-ten blocks to show decomposing a ten. Students build on this understanding as they use base-ten blocks to represent the starting number and subtract an amount that requires them to decompose a ten.

• “What did you need to do with the blocks when Diego took away 8 ones?”
• Select previously identified students to share.
• “What is different about using base-ten blocks compared to the towers of 10?” (With the towers we could break it apart into ones, but with the base-ten blocks we had to trade a ten to get 10 ones.)
• “When you subtract by place, sometimes you need to decompose a ten to subtract ones. When we use connecting cubes, we can show those by breaking apart or decomposing a tower of ten.”
• “When we use base-ten blocks, we can’t break apart the ten into ones, but we can still show decomposing it by replacing 1 ten with 10 ones.”