Interpreting remainders is something the 4th grade teachers and I talk about a lot because so many students seem to struggle with it. Students can typically determine if they need to divide and find a way to get the answer, but if the remainder impacts the response it becomes more difficult. I believe the struggle is not so much about the remainder, but more about students making sense of problems. Many students love to compute the numbers in the problem and get an answer quickly, however they rarely revisit the problem to see if their answer makes sense. I found an even more interesting thing in their work today though that left me thinking about how their solution path impacted the way they dealt with the remainder.

I launched the lesson with a story. If you read my post on numberless word problems, this will be very familiar. I posed the following story to students and asked what they noticed and wondered:

*Mrs. Gannon is having a picnic and inviting some people. She is going to the grocery store to buy bottles of water and packs of hot dog rolls. *

Since the students were on the carpet in front of the SMARTBoard and there was not much space to stand and write on the board without trampling a kid, I decided to sit off to the side and type their responses.

After they shared the things they noticed and wondered (in black font below). I told them I would give them some information that would answer some of the things they wondered. After typing in the numbers, I asked if what they noticed and wondered now (typed in red font below).

(Side note: The cost of things is something I would love to weave into a lesson in the future because that came up a lot and will be great to see what they do with some decimals.)

Since they noticed how much water Mrs. Gannon needed, I wanted to see how they dealt with the hot dog rolls because the remainder would make a difference in the answer. I asked them to find how many packs of hotdog rolls she would need.

Some divided and got 4 r 4 as their answer (the skip counting on these pages came after their chatted with their group.

Some skip counted to get the packs of water and hot dogs:

Others used some multiplying up, some right, some interestingly not:

While I could probably talk for a while about all of the interesting things they did in solving the problem, the most interesting thing to me today was looking at who got the correct answer of 5 packs on their first try.

This is what I noticed as I walked around:

- The students who went straight to dividing said their answer was 4 remainder 4, no reference to the context, no mention of using that remainder for anything.
- The students who skip counted nailed it on the first try. They said as they counted they knew 32 rolls were not enough for everyone so they needed to keep going to 40 so everyone got one. They mentioned the context throughout their entire explanation.

I continued the conversation with Erin, the reading specialist, when I got back to the room. We started talking about how this contrast could play out in two different scenarios:

- On a standardized test, given this same context and answer choices of 4 and 5, the students who could efficiently divide may choose 4, while the skip counter would have gotten it correct.
- On the same test, give a naked division problem, no context, the efficient divider gets it correct but what about the skip counter. Can they think about the problem the same when their is no context or does skip counting make most sense with a context?

Because I thought it would be interesting, before I left, I asked them how many people she could invite so she had no leftovers at all. Fun stuff to end the class…

And of course, some students are just funny….

Joe SchwartzLove the opening move to a numberless word problem. This reminds me of this lesson from Robert, which I’ve never used but have always wanted to try:

http://robertkaplinsky.com/work/how-many-hot-dogs-and-buns-should-he-buy/

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