# 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

# A Great Day of Decimal Division

Today, I really saw such a beautiful picture of the culture of learning in my class and marveled in the way in which my students had arguments in the best possible sense of the word. They were excited about the math, working so hard on proving their answer, and in the end ok with being wrong because they “saw where they messed up.”

I posed the the question 2 ÷ 1 on the board and asked the students to write what they thought about when solving that question, or any division equation similar to that. We shared and collected our responses…

Going back to their journals, there were some other interesting ones such as contexts and what the symbols meant…

I then asked them to write this equation and answer they thought for 2 ÷ 0.1. I was so excited to have a split class of the answers 20 and 0.2. They worked on proofs with their tables and I got some great thoughts around what the think about division as well as references to visuals they think about in their solution.

I particularly like this answer because of the way it was written…

At this point they begged for chart paper to create a poster to “show the other group why they are right.”

Each group shared their thoughts and there were a lot of “Oh”‘s and “I have a question for…”‘s going around the carpet area. At that moment, they were completely owning the class. They politely waited to ask their questions, politely disagreed with one another, and openly admitted when they “changed their mind.” I loved this moment so much, I just listened.

I had all but one student who still did not agree yet with the 20 so I had him write down why so I can think about how to structure tomorrow to meet his concerns. He is focused on adjusting either the dividend and/or divisor and then adjusting the quotient. I love all of the “how”‘s.

I asked the ones why had 20 to take a stab at a context for the problem and others who changed their mind to tell me what part was the final aha for them.

Today was the day I wanted to have other people in there watching to be as excited as I was. I told the kiddos how proud I was of them and off to lunch they went! I then had to bottle up my excitement until I could get this all out. It was just a really great day to watch my class work, and learn, together.

-Kristin

# Multiplying Decimals Less Than 1 Whole

Apologize, not much time to write, but today was so cool I had to share!  I am in the midst of using this work to better plan for tomorrow.

Short version: After our predictions yesterday, I posed 0.4 x 0.2 (I changed the problem to 0.6 x 0.4 for my second class) and asked the students to individually jot down what they thought the answer is. I was looking to see how they intuitively thought about the problem.  As expected, I saw 0.8 and 0.08 (2.4 and .24) as I walked around. I wrote both answers on the board, asked them to write their reasoning in their journals and then we shared as a class. No telling which was right or wrong, just sharing and listening.

Some great thinking and critiquing of each others’ reasoning ensued and then I sent them off to come to a consensus as a table and create a poster of how they thought about it!

Now, where to go with this work? They could just look at strategies, but I want them to think deeper about the meaning of the problem.  After chatting with my colleague Faith tonight, who is coming to observe tomorrow, we are going to have the students walk around to the other group posters and talk about what they saw on the other posters that changed the way they thought about the problem.

From there, I really wanted them to think about a context for this problem and Faith suggested also thinking about what happens are you begin to adjust the numbers and why….really thinking about the reasonableness of answers. What happens when one factor increases? What happens when one factor goes over a whole? What happens if the factors go into the hundredths? Does the product increase or decrease? Why?

So many fun convos to be had tomorrow!

-Kristin

# Decimal Multiplication: Whole # x Decimal

Through numerous Decimal Number Talks, Investigations on tenths, hundredths, and thousandths grids, and many findings about decimal operations, we are approaching our last couple lessons in our decimal unit. Not that the work with decimals ever ends, but our unit ends with decimal times decimal and the generalization of a “rule” for multiplying decimals. I have many thoughts about the new Investigations unit on multiplication of decimals but I am very excited about the connections my students have made between whole number and decimal operations. I do attribute a lot of their flexibility to our Number Talks though:)

I wanted to assess where they were before we moved into a decimal times decimal work because I think there is a lot of reasoning to do there before we come to a generalization!  I was really excited to see the use of multiple strategies!

First, I had students who were still treating the decimal operations like whole number operations and reasoning about where the decimal point “makes sense.” I do love this because it is heavy in estimation and sense making about what is reasonable. It is obviously not the most efficient strategy, but I what I truly learned from this, is that I need to do more whole number multiplication work with this student to build efficiency…

I have students that love partial products….(and I cannot get some students to stop saying the “box method”….:)

I loved this area model because of the size of the .4 side. She was very particular about that!

Some friendly number work…I especially loved her estimation first….yeah!

I had some who multiplied the decimal by 10 and then divided their product by 10…

Saw some halving and doubling…

I had a student think about the decimal as a fraction. It starts at the top and then he jumps to the bottom of the page.He said he multiplied 9 x 12 to find out how many “rows” he would have, 108. Then he divided it by ten because there were 10 rows in each grid.  It was interesting!

So tomorrow we start decimal by decimal multiplication…I feel great about our start and I look forward to having them reason about decimals less than a whole times less than a whole.

-Kristin