# Remainders: Division & The School Year

Looking ahead in planning for the remainder of the school year, I am currently finishing up my decimal unit and excited to end the year with Growth Patterns. I was planning on finishing decimals this week, however, I have one more thing that I feel is missing from the unit that I am curious to see how students are thinking around it. In the unit, within decimal division, the students are very comfortable estimating quotients and thinking about a variety of strategies in finding how many of the divisor are in the dividend. However, one thing that is not addressed is remainders, and how we notate them. I had not really thought much about it because in the context of the problems we were doing, the remainder made sense. However, during a division number talk, not within a context, the “r” was still there. It bothered me a bit. When I asked how they could write the quotient as a number, I got blank stares. I know fourth grade really spends a lot of time on interpreting remainders, but do we spend equal time on various notations of the quotient?

I have decided to extend my decimal unit just a bit longer because I feel this is something my students can definitely reason about and I am curious the connections they can make between whole numbers, fractions, and decimals. I decided to start with whole number divisors and dividends and move to decimals from there. Today, I gave them the problem 256 ÷ 20. They estimated somewhere just over ten and then I asked them to solve it. If they finished early, I asked them to write a context to match the problem.

The majority of the class’ work looked like these and contexts involved a sharing situation…

When pushed to write their quotient as a number without the “r,” most said this…(I do love the way this student divided:)

I did get a few 12.8 and 12 16/20, which interestingly fell more in money contexts…

All of these, I had anticipated, but then I got some really great unexpected answers that allowed students to think about the connections between notations…

12.5 r 6         12 16/256         12.75 r 1

I wrote these responses on the board and asked the students to see if any of the answers meant the same as 12.8 or 12 16/20 or 12 r 16, that we had established were the same. They did also mention, which I loved, that certain situations my use different notations.

I had some amazing proofs that we are kicking off the day with tomorrow before moving into decimal divisors. While I was hoping for students to look for equivalencies in the quotients themselves, most groups went back to trying out division in a different way to prove the answers. This group went back and solved the problem using the same method every time, just changing the breakdown of the quotient.

This group used multiplying up to see that 12.5 r 6 worked as a correct answer.

After asking them if they saw any relationship between the quotients, I got this…(much more what I was hoping to see in their reasoning)

This student is still sticking with 12 16/256 and quite honestly I don’t know how to approach this one. It is a different way of writing the remainder and I cannot decide if there is a time when this would be an appropriate notation?

The most perplexing quotient for most of the students was the 12.75 r 1 so I asked the student to write out his thought process because he was having trouble explaining it.

Now, while the entire class period seemed to focus on the remainder in a division problem, this explanation represents the remainder of the school year! I asked the above student to go in the hallway and record his thinking through the problem because he had such a beautiful way of starting to explain how he decided how much to add based on the distance from the dividend…but then I got this 🙂 https://www.educreations.com/lesson/view/kewl-aid/31841872/

And here’s to the remainder of the school year….

-Kristin

# Decimal Division, Running & Why I Love My Tweeps

Yesterday, I posed a decimal division problem to get my students thinking about what division means to them and how that applies to decimals: https://mathmindsblog.wordpress.com/2015/05/05/a-great-day-of-decimal-division/ (It was a really great day)

I was thinking of moving into a context today to see how they would represent the problem and the approach they would take after yesterday’s discussion. So, of course I threw it out on Twitter…

All evening I was thinking about a context and this one Elham suggested worked great for me! I was still thinking about how to word it to be something that the students may be connected to, then Joe’s tweet came this morning after my run…

Duh, my runs! Thank goodness Joe was up early too!  My students know I run every morning and cannot fathom that anyone actually wakes at 4:30 in the morning, so I knew they would love this.

To start the class, I posed..

“I ran 2 miles on Monday afternoon. Every .4 mile I took a sip from my water bottle.How many sips of water did I take during my run?”

As with most times, I gave them some individual time before consulting with their table mates. It was awesome to see so many of the connections to yesterday’s work and also new representations that did not show up yesterday.

This one was so interesting how he broke up the mile to .4 +.4 +.2 and then combined the .2’s to make 5 four tenths.

This number line was so nice and then I loved how he got to the end and then counted the jumps going back down to zero. Also, at the top he had multiplied up to the 2 miles, nice way to show two ways of thinking about the problem.

There was a lot of skip counting by .4, but this model was especially wonderful. It is an area model combined with a number line. He counted up by .4 in squares that attached until he reached 2. I would expect students to count the number of .4 sections to find the answer, however this one labeled the 1, 2, 3, 4, and 5 at the end of each section.

I then gave them a log of my past five runs. I told them to assume that I still take a sip of water every .4 mile. I wanted to know how many sips I took and then how much further I had to go until my next sip.

I got some awesome partial quotients, number lines and multiplying up.

Now, the conversation of remainders came up. They want to know how to write the answer without the “r.” They wanted to know if they could write that as part of the number that was the answer. For example, could they write “7 sips r .2 as 7 1/2?” Saving that for tomorrow.

And THIS is why I love the #mtbos….my lessons take wonderful twists that make the learning experiences in my classroom so much better for my students! No teacher can do this job alone!

-Kristin