Practice Puzzles: Math Practice Two

Math Practice One and explanation of the puzzles can be found here.
 

Math Practice Two: Reason Abstractly and Quantitatively
This practice is when a mathematician makes sense of the problem in context, then ignores the context to do the calculation, then steps back into the context to make sure the answer makes sense. For example, a child may make sense of the following math story as a combining story:
          "There are 6 kids playing soccer. Some more kids come and now there are 10 kids playing soccer. How many kids came to join the soccer game?"
        In other words, they see this as combining the kids who were playing with the kids who came to join them, and ending up with 10 kids. The model is 6 + ____ = 10. Even though this is additive in context, a child may solve the problem by subtracting 10 - 6 = ____. Once they've made sense of a problem, they can solve it however they want. Then go back and see if it still makes sense.

 I watched my students do this a lot the first few weeks of second grade. They would build the two parts (contextualize) then put them together and count them by 1s, 2s, 5s....however they wanted (decontextualize). They started out building everything as single blocks, and then combined the quantities into one long train. (See the picture above) They were able to go back to the context and tell me the unit, for example that the answer was a number of stickers for this particular problem. But once they made the long train, the original parts were lost to them.

Look, here was the problem: "Diva had 4 stickers. She went to the store and got another 8 stickers. How many stickers does she have now?" So the problem was that they could answer (12 stickers)...but if I then asked, "So how many stickers did she buy?" they would say "12". Because they totally forgot about the parts once the parts were swallowed up into the whole amount that was the sum.

This is problematic because keeping track of those parts is what connects addition to subtraction and it's what is going to allow us to solve subtraction problems as missing-part addition problems (for example, seeing 12 - 8 = _____ and thinking 8 + ___ = 12).

As they got better at thinking about quantities, I started making a VERY big deal about students I saw who were making the parts and color-coding them or counting by 10s and then continuing with the ones without physically moving the cubes to be together, (See picture above) thus maintaining the two parts. When I asked them I would usually say, "Which ones are the stickers she had? Which ones are the stickers she bought?" and when they could answer it was a very good thing. I'd even reserve their trays intact, to use as a mini lesson to start our next workshop....."Friends, do you see what she did??? Like all mathematicians, she arranged this so we could tell which ones were the stickers she bought! Can we answer the question of how many stickers she has now? Of course we can! AND we can still see the two numbers she made....Who remembers what this first number means? What about this second one? And how many all together? Who thinks they can work on keeping their parts like Ariel does? Off you go."