• Making the Shifts
• Task
• While some students might try to simply add, others will recognize that $23$ ones is $2$ tens and $3$ ones. When we combine the $2$ tens with the $8$ tens we already have we get $10$ tens, which is one hundred. So we have $3$ hundreds and another hundred and three ones, which is $403$.

  Commentary

  The purpose of this task is to help students understand composing and decomposing ones, tens and hundreds. The task is meant to be used in an instructional setting and would only be appropriate to use if students actually have base-ten blocks on hand. The last two tasks fully engage the notion of composing and decomposing as needed for algorithms for addition and subtraction. Both parts require persistence, as in the Standard for Mathematical Practice 1.

  After seeing the first two tasks, students have the ideas needed to start listing possibilities in the third task. the idea of exchanging a ten for ten ones and a hundred for ten tens is needed in order to complete the task.

  Solution:

  While some students might try to simply add, others will recognize that $23$ ones is $2$ tens and $3$ ones. When we combine the $2$ tens with the $8$ tens we already have we get $10$ tens, which is one hundred. So we have $3$ hundreds and another hundred and three ones, which is $403$.

  • Lamar could use ten ones for each ten-block which he was missing. So instead of $2$ hundreds, $6$ tens and $1$ one as he wanted, he can start with the $2$ hundreds and $4$ tens which he has and then use two sets of ten ones instead of the two more needed tens. Those make $20$ ones, which we add to the $1$ one needed to get $21$ ones. Collecting all of these we get $2$ hundreds, $4$ tens and $21$ ones. There are many possible solutions – for example using $2$ hundreds, $3$ tens and $31$ ones – but the one given is the most likely.

  The list of all ways using $1$ hundred is:
  

  • $1$ hundred, $2$ tens, $4$ ones.
  • $1$ hundred, $1$ ten, $14$ ones.
  • $1$ hundred, $0$ tens, $24$ ones.

  The list of all ways not using any hundreds is:

  • $12$ tens, $4$ ones.
  • $11$ tens, $14$ ones.
  • $10$ tens, $24$ ones.
  • $9$ tens, $34$ ones.
  • $8$ tens, $44$ ones.
  • $7$ tens, $54$ ones.
  • $6$ tens, $64$ ones.
  • $5$ tens, $74$ ones.
  • $4$ tens, $84$ ones.
  • $3$ tens, $94$ ones.
  • $2$ tens, $104$ ones.
  • $1$ tens, $114$ ones.
  • $124$ ones

  To know the list is complete as we make it, we can start with the standard way, namely $1$ hundred, $2$ tens, and $4$ ones, and exchange tens for ones one at a time to get the first list. Then we exchange the hundred for $10$ tens, to get a total of $12$ tens along with $4$ ones. Once again, we can exchange tens for $10$ ones step by step in order to get the second list. Because we cannot use two or more hundreds, these two lists contain all possibilities. 

• Additional Thoughts

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