A B C

Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6

Problem B1

 a. The Digi-Blocks forced students to group in tens and hundreds. Most students started with the larger number of blocks. Students then took apart tens to obtain more ones, or hundreds to obtain more tens, as they completed the subtraction processes. b. These blocks are automatically grouped in tens and hundreds. Students "see" the connections between the place value and the digits they are writing. c. In this case, most students started with the larger collection and took away the smaller collection.

 Problem B2 With this manipulative, the symbols students write on the paper match the actions they took with the blocks.

 Problem B3 Ms. Weiss's lesson is structured around understanding the meaning of operations, in particular subtraction and addition. The students utilize a variety of methods and tools to solve problems. They also use multiple manipulatives, such as Digi-Blocks, which are particularly helpful in strengthening students' ability to compute fluently and understand place value. The students using Digi-Blocks are also learning to count with understanding and recognize "how many" in sets of objects.

 Problem B4 Ask students to talk about what they did with the blocks and what they wrote down on their papers. Listen for statements like this one: "I needed to take away one of the ones, and I didn't have any ones. So I had to take apart one of the tens." Students should be talking about taking apart and putting together units of ten.

Problem B5

 a. Reed understands the concept and has good mental math skills, so he can do the computation in his head. He is well able to think of shortcuts to make mental computation easier. Leland also has a good understanding of the concept and has the mental math skills to think of shortcuts for mental computation. Randy does not understand the paper-pencil algorithm for subtraction with regrouping. He heard both Reed and Leland discussing their answers, but it didn't seem to bother him that his answer didn't match theirs. b. Reed is probably ready to work on and understand other methods. For example, he could work on a written algorithm to match his informal understanding. We do not know how well he could do this with paper and pencil. Leland is probably ready to work on and understand other methods. He also needs to work on a written algorithm to match his informal understanding. You can test his ability to do this type of computation with paper and pencil. Randy is probably not ready to move on to new methods. He needs to work on some mental math skills and sense-making for subtraction with regrouping.

Problem B6

 a. Mariko has entirely misunderstood the problem. She has found the sum of the two numbers rather than their difference. This may indicate that she is using key words without really reading the problem. She might have interpreted the key word "more" to mean addition. Tarra counted up by tens, starting from 35, until she got close to the desired number, and then counted backward by ones until she reached the goal. She knew to count backward instead of forward here because she had passed the desired number, 52. This is sophisticated thinking. She appears to have a good understanding of the concepts and procedures. Daniel counted backward by tens until he got close to the number, and then counted up by ones until he reached the desired goal. Daniel thinks that since he is now at 32, he must count up three to get to 35. He does not realize that he passed 35 on the way to 32. b. Mariko is probably not ready to move on to new methods. She needs some hands-on practice with subtraction interpretations with smaller numbers. She does not appear to know that comparison requires subtraction. She should be discouraged from choosing an operation based solely on key words and encouraged to read and reread until the problem makes sense to her. Tarra has a good understanding and is probably ready to move on to new methods. She now needs to be given more complicated problems, to test whether she can do subtraction using paper and pencil. For Daniel, new methods may or may not be helpful. Although he understands the concept of subtraction, he still needs to practice with smaller numbers to strengthen his procedures for mental subtraction.