Listen. This is the sound of sense making.

I taped this audio recording at the end of the day.   The three boys were waiting to picked up. They came over and said, “What are you doing Mrs. Caban?” I told them I was reading a blog post about decimal quick images (thanks Kristin Gray).  They asked me if they could do some more problems like we did this morning. So, I showed them the one above.  It was so cool to listen to their voices collectively buzz as they wrestled with unit conversion.

I started the morning with these boys in their fifth grade math class.  Mrs. Gordon and I have been co-planning and co-teaching this week. We are introducing operations with decimals and we decided we want to anchor the unit in the student’s prior knowledge about decimal fractions.

Yesterday, Mrs. Gordon did count around the circle. First, they counted by tenths. We had planned that she would intentionally ask each student how he/she wanted Mrs. Gordon to record what they said – at least in the beginning. We were curious about how many of them were actually picturing decimal fractions and how many were picturing decimals. We knew that how we recorded what a student said would influence how the subsequent students in the counting sequence responded. We also decided she would record in rows of 10.  Some interesting things happened.

The boy who started the circle told Mrs.Gordon to write “one over ten”.  The tenth person said, “ten tenths or 1 whole.”  Mrs. Gordon said the students seemed to enjoy the challenge of changing from mixed numbers to decimals to improper fractions and back to decimals again.  They noticed that the halfway point of each line was always composed of 5 tenths.

Then they counted by hundredths.

Today, when I went in, we wanted to see if we could get the kids to establish some relationships between tenths and hundredths.  We knew we wanted to try to switch the unit as we counted – start with tenths, move to hundredths, move back to tenths. We decided to record the counting sequence on a series of blank number lines that were segmented into ten sections.

When we got to 1 and 1 tenths, I asked everyone to pause.  I told them we were going to switch to counting by hundredths. I asked Liz if she wanted any suggestions from her classmates or if she wanted to try it herself.

Liz said, “Well. I think it will be 1 and 1 tenth and 1 hundredth.”

We wondered, “What do we call that?”

“Well,” said Liz “I think there are ten hundredths in 1 tenth so that would be like having 1 and ten hundredths plus another hundredth which is 1 and 11 hundredths?”

I wish I had written 1.10 underneath 1.1 on the recording sheet,  as Liz was speaking, so she could see the ten hundredths that she had so elegantly described.  We continued to count by hundredths until we got to one and 17 hundredths. Then, I switched us back to tenths. There were audible gasps.

“Whoa.”

At this point, we could tell that the transition back to tenths would be tricky for a good number of the kids. There were some who thought they could just add one tenth to the tenths place, but they struggled to convince some of their peers.  One student was stuck.  I asked him, “Do you think you could use a number line?”

He drew a number line and figured out that he could decompose the tenths into ten hundredths.  Then, he broke up the ten hundredths into 3 hundredths – to get the nearest tenth and then added 7 hundredths to arrive at 1.27.  Yay! We were really hoping someone would consider decomposing.  He still had to count be one hundredths to arrive at his solution, but he was able to see the .3 and .7 chunks after he counted. I decided to add his representation to our chart. You can see it above.

I wish that I used a bigger number line so everyone could see the hundredths. I also wish we grouped the kids in smaller circles so everyone could engage in more counting. So… we made three big number lines to use later in the week.