#### Case 6: Double Compare

The students in this study have come to the conclusion that a number added to a larger number is greater than the same number added to a smaller number. When Martina had 6 and 2 and Karen had 6 and 1, Karen quickly said “You.” I asked how she knew and she pointed to the 2 and said, “This is big. Even though these are the same (the 6s), this (the 6 and 2) must be more.” (Page 31)

The second generalization the students formulated was that the sum of two smaller numbers is less than the sum of two larger numbers. Lola displayed a couple examples the groups came across during the game, and asked how they knew which player won that hand. Each student went on to explain that when both numbers are larger, then that person wins. They came to this conclusion because they simply “just knew” it was correct.

#### Case 7: Adding 1 to an addend

According to Joe on page 38, “Adding one more to the number you had—to any number you pick—it changes one higher than the other one, than the other number: 10+1- and 10+11. The 10 changes one higher.” The students were able to comprehend the solution to doubles and knew what happened once 1 was added to an addend. As Joe stated above, they all agreed by adding one, the solution will increase by 1. They knew this was also true if you added any number, the solution would then increase by that amount. Maureen led her class into thinking about other possibilities, such as non-doubles. Connie volunteered and shared with the class 51+53 and 51+54. Students used the same rules they knew applied to the doubles in order to solve these equations.

#### Case 8: Was it something I said?

Most of the students were able to give an example of taking away from and adding to numbers. (99+1=100; 98+2=100; 97+3=100 and then 96+4=100) The teacher asked for someone to explain what’s going on in the set of number sentences, Molly raised her hand. She explained that “..1 number less than the 100 and one number more than zero” is how you begin. A few minutes later Scott demonstrates the same concept through the use of cubes. Scott used a stack of ten cubes to demonstrate 100. First he took one off the stack and moved it into a second stack and continued to do this, showing one stack getting smaller and the other getting larger. Molly then states that you can always change around the order of a problem to make it easier to solve. She goes on to mention that addition is two groups of individual items being put together creating one group of items. During this case the students are clearly displaying strategies that make their computation more efficient.

#### Case 9: Is it two more or two less?

The Generalization Max has made about subtraction is foundon page 47, “the less you subtract, the more you end up with, and in fact the thing you end up with is exactly as much larger as the amount less that you subtracted.” If you have 145 pennies, and 100 are taken, you are left with 45 pennies. 145 -100 = 45

#### Case 10: 37 + 16 =40 + x

### QUESTIONS:

**1.Consider the passage from lines 450 to 475 of Monica’s case (Case 9).**

**a) What is the generalization that Monica’s student, Brian, articulated? Did he prove it? If so, how? How did Rebecca and Max elaborate and refine Brian’s idea? – **Brian articulated that when you take a piece away from a whole thing and have some part left over, then you start again with the same size and take a smaller part away, and you will make more left over than the first time. He justified his idea through a labeled representation. Rebecca helped by mentioning if it were a loaf of bread and it would be similar if you took out different size bites or pieces to relate Brian’s to real life. In mathematical terms, Max discusses the less you subtract then the more you’ll end up with. He also states that you can find the amount you end up with by adding it to the first amount the number less than you are subtracting by.

**b) Once you have thought about the mathematics of this passage, turn your attention to the dynamics of this classroom. What do you notice about the ways students and teacher interact? – **This classroom has a welcoming and positive vibe where all students have the chance to voice their understandings and predictions. The teacher begins each topic discussion and allows the students to chime in with what they know, and allows them to add on to other student’s comments.

**2. In Case 8, the teacher, Kate, says at line 342 that her students “had learned something important about the operation of addition.”**

**a) Consider students’ discussion from line 286 through line 340. Identify the flow of the mathematical ideas in this segment. What learning did the students reveal? – **The students in this section revealed that if you have x + 0 is is equal to (x-1) + (0+1).

**b) How did the mathematics students were working on in that passage, lines 286 to 340 show up in the computation later in the case? – **Through using the same generalization, they applied it to finding an equal addition equation for two other addition equations. You can do this by subtracting or adding a number from the first number and then doing the opposite with the second number. Creating an always equal solution.

**c) Consider Molly’s comment at line 394. What question was Molly responding to? What was significant about what Molly said? How did Kate follow up?- **Molly claims you can always change the order of a problem to make it easier to solve. She goes on to mention that addition is two groups of individual items being put together creating one group of items. Kate follows up by asking “is this always true? Does this straticy always work?” The teacher then responds by relating back to the materials the class had been working with.

**3. What is the main idea that students in Maureen’s class (Case 7) were working on? What did Sema demonstrate with cubes (line 101) and how did Connie and Ester connect with it? – **The main idea students are working on is how does the solution change if you add to an addedn. Sema demonstrates that adding one cube to an addend makes the solution one greater. Connie and Ester claim that when you add one more to the addends, it adds one more to the solution.

**4. Examine the class discussion in Carl’s case from lines 603 to 628. Explain the thinking of these students. What images of the operation do they call upon in their work? How is the generalization the middle-school students are discussing the same or different from the generalization implied in the work of second graders in Case 8? –**This discussion is based on thinking of addition equations and making equivalent addition equations and manipulating two piles. No matter what you add to one pile and take away from another pile, the total items you have remains the same. “It is like balancing and what ever you take away you have to add back or else the problem is not the same,” which is a generalization from the middle schooled student. The middle school students are able to view this with more steps than then second graders.

**5. What is the issue that Tammy brings up in line 635 of Carl’s case? Does the generalization you wrote for this case still apply in the instances Tammy brings up? Explain why or why not.- **Tammy’s issue works because even though changing the 27 in 27+3 to 30, requires you to make the 2^{nd} addend zero by subtracting 3, which still works because 30+0=30; still equaling 27+3. This strategy can work for decimals as well as long as you are sticking with addition and subtraction.