The purpose of this activity is for students to use their knowledge of base-ten diagrams and place value to make sense of a subtraction algorithm. Students notice that in both the base-ten drawing and the algorithm, the subtraction happens by place. We can find the difference of two numbers by subtracting ones from ones, tens from tens, and hundreds from hundreds, and adding these partial differences to find the overall difference.

Students also recall that sometimes a place-value unit needs to be decomposed before subtracting. For example, a ten may need to be decomposed into 10 ones first. This decomposition can be seen in both the base-ten drawing and in the algorithm. In the Activity Synthesis, students interpret the work and the reasoning of others (MP3).

In the Activity Synthesis, record a completed version of the written algorithm that includes parentheses around each expanded form expression. Although students should not be required to use parentheses in this way when using this algorithm or other similar student-invented algorithms, this notation helps make the subtraction more clear and helps build foundations for the properties of operations.

• Invite students to share their responses.
• “How did Kiran know to rewrite 391?” (If he tried to subtract the ones, he would notice he doesn’t have enough, so he needs to decompose a ten into 10 ones.)
• “Why is he able to rewrite as ?” (Both expressions add up to 391. Each shows 391 in a different way.)
• “How is Kiran's reasoning like Jada's reasoning?” (Both decomposed the numbers into hundreds, tens, and ones, and subtracted the numbers in each place separately. Both decomposed a ten into 10 ones.)
• “How is their reasoning different?” (Jada used base-ten drawings to represent the numbers, and Kiran wrote them in expanded form.)
• “How did you finish Kiran’s work?” (I subtracted the ones, the tens, and then the hundreds. I subtracted the hundreds, the tens, and then the ones.)
• Record Kiran’s completed algorithm, and keep it posted throughout this lesson and the following lesson:

• Highlight the following points while recording the completed algorithm:

  • Kiran decomposed each number using expanded form (enclose each expression in parentheses to emphasize this point).
  • Kiran subtracts ones from ones, tens from tens, and hundreds from hundreds.
  • He shows decomposition by crossing out the original tens and ones and replacing them with how many he is taking away or adding to each place.
  • Parentheses help serve as a reminder that the subtraction still involves 2 three-digit numbers when using this algorithm.

The purpose of this activity is for students to analyze the connections between base-ten diagrams that represent subtraction and algorithms. In particular, students relate how the two approaches show a hundred decomposed into tens, and a ten decomposed into ones, in order to facilitate subtraction.

As students work, encourage them to refine their descriptions of what is happening in both the diagrams and the algorithms, using more precise language and mathematical terms (MP6).

• Invite 2 or 3 students to share a match they made and how they know the cards go together.
• “In the diagrams, why was a hundred sometimes moved over into the tens place?” (This happened when there weren’t enough tens to subtract. We used a hundred to get more tens.)
• “How was this shown in the algorithm?” (A hundred was crossed out and 10 tens were added to the tens place, like in D—20 was crossed out and 120 was written above it.)