**Case 8 – Bunnies and Eggs**

The bunny problems were incredible …. I couldn’t believe that kindergartners could figure that out so easily. Using creativity with their manipulatives such as keys and blocks was very resourceful. All the children, Jason, Rashad, Carlita, Kenya, and Flora all put together some sort of three groups of four – some did count them all up, but ultilmately, they were all using multiplication strategies. Jason used blocks, Rashad and Carlita used keys, Kenya used Unifix cubes, and Flora used rocks – all using a different manipulative to help them solve the problem. Junior was making similar groupings but had trouble counting each component of the group and instead kept counting groups. He did not see the big picture that each group represented a basket of smaller items.

**Case 9 – Easy Multiplication**

In this case, 2nd graders were given an incomplete rectangle to work with but they did very well with it. They saw very quickly that there were 5 groups of 3. But, interesting that Christopher wrote that as 3+3+3+3+3 instead of counting by 5’s. Then Patrick identified a different way of seeing the groups as 3 times 4 plus 3. I noticed that Nisha would announce the different thinking processes to the class – this gives students other ways to understand and learn. Just like when we ask several different students to come up to the board to write how they solved a problem – we can see different thinking strategies and are thus exposed to new ideas. This information sharing was also done when she asked them to share their strategies in small groups. Further, visual displays were used and recorded to help understanding and strategy sharing even more. Then, that was when the students started connecting their ideas to other ideas that they had heard – WOW! It was noted that the students used lots of repeated addition to accomplish their multiplication.

**Case 10 – Candy Canes in Packages**

Janine’s group struggled to make send of buying candy canes for 609 students at the price of $0.29 for every package of 6 . One of the girls, Letitia, struggled to keep up with adding groups instead of single numbers. Janine kept seeing her understanding but then it would fly away out of her head and she would be confused again. This was similar to Junior’s problem in that they couldn’t keep the idea of “groups” in their head and could only “see” single items instead of things together.

**Case 11 – How Do Kids Think about Division?**

Vanessa is considering division in the traditional way in the first problem. But, in the second problem when she is asked to figure out how many packs of seltzer water to buy if they come in packs of 6 and they need 36 total, she is using the logical inverse of division which is multiplication and really addition to figure how many times she will need to add 6 together to get 36. It really show me that she has a great understanding of the math concepts and is not worried about using the “correct” operation.

Next, Cory is using his understanding of working with more familiar numbers first and is trying to use 2 first, then use 2 again. That show me really incredible understanding of what he is doing. So, he made a mistake, but ultimately, he is trying to divide 36 by 2, then divide 18 by 2, and he comes up with 9 – WOW! I don’t think I would worry that Matthew is figuring his division problem using multiplication. If he can see the connection and work successfully, it will only be a matter of time until he can clearly see how division is separate (but still connected to multiplication).