Notice and Wonder: Division and Easter egg math

Some of the best math is the math that happens organically, math born out of a need to answer a real-world question that has importance to the mathematician. Since Spring has officially sprung, naturally the meaningful math that is happening at our house revolves around plastic Easter eggs and the treats that go inside. The problem solving began with these materials below:
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My daughter is supposed to bring 12 filled Easter eggs for the Kindergarten Easter egg hunt this week, so the question that we wanted to answer was: If we want each egg to have the same amount of Smarties, how many packages of smarties will each egg get? Two powerful questions guided our conversation:
I asked Cora what she noticed. "I noticed that there are 12 eggs because there are two bags of 6, and I noticed there are a lot of Smarties!" I then asked if she thought there were more or less than 12 packages. She thought about it for a brief moment and said with confidence, "Definitely more than 12." I shared with Cora what I was wondering: "I wonder how many packages of smarties each egg should get." Cora piped up with, "It has to be fair!"

Division in the Common Core State Standards first appears in 3rd grade in the Operations and Algebraic Thinking domain when students partition a number of objects into equal shares. However, an intuitive notion of "fair shares" appears much earlier than 3rd grade for most students. All you need to do is ask a 5 or 6 year old to share something - they'll let you know if it was fair! 

Consider for a moment what strategy you would use to approach our Easter egg/Smartie problem. It is most likely one of these:
  • Count all the Smarties, divide the number by 12, put that number of Smarties in each egg
  • Distribute the Smarties one at a time into the 12 eggs until all the Smarties are gone, checking to make sure you have shared them "fairly" or equally
Cora went with the second strategy:
Distribution of Smarties, one Easter egg at a time
Once each egg had one package of Smarties in it, she began to put one more package in the first egg. I expected her to continue as she had the first round, distributing the remaining packages to the eggs, but just as the students in my classroom always surprised me with their thinking, so did she. Once she had put a second package in the first egg, she closed the egg. 

I asked her why she had closed the egg. "Because I know it's not going to get more than two," she said matter-of-factly. I inquired further. "Well, I noticed that there were enough Smarties for the eggs to have two, so if I put three in this one, it wouldn't be fair, so I closed it." Oh, the power of estimation! She was so confident with her "noticing" about the remaining Smarties, that she closed shop on distributing any more packages to the first egg! She continued the process by closing each egg after she had distributed another packages of Smarties. She got down to four eggs and exclaimed with delight, "I knew it! Well, I was wondering!"
Four eggs left to get another package of Smarties, and four Smarties left!
Filled eggs, ready to go
Why is math so awesome!? I wasn't quite finished... "So, I wonder how many packages of Smarties we had to begin with." Cora looked at the eggs. "I think I'll count by two's since each one has two." Solid strategy! 2,4,6,8,10,12,14,16,18,20,22,24... "24! Too bad there wasn't 25. I wanted one to be leftover so I could eat it." A mathematician after my own heart. 

Now, I notice that our task is complete, and I wonder if we can hide the eggs long enough for all of them to make it to school with two packages of Smarties still inside... 


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