The first day back to school after NCSM/NCTM is definitely an exciting one! I was excited to see my students, hear and see them doing math again, and incorporate the amazing things I learned at the conference with them. It is always great when I can go to a session, regardless of the grade level focus, and be curious how my students would engage in the activity. For example, I went to an amazing session on counting by Kassia (kassiaowedekind), Elham (@ekazemi) and Allison (@allisonhintz124). While the session focused on whole numbers, I began thinking about how I could take this same practice of Choral Counting and use it in my classroom. I have to admit, while my first thought was what my students would think about during this activity, I also had my own curiosities in the teacher organization of the work. Does writing them horizontally vs vertically bring out different noticings or patterns? or How does how many I put in each row or column affect their thinking about it?
Luckily, we are currently working on multiplication of decimals and I thought this would fit in just perfectly. I did some brainstorming and decided for my first class we were going to choral count by 0.3, record horizontally and vertically and have 1/2 of the class focus on the horizontal while the other 1/2 focused on the vertical. I was curious to see if they saw different patterns emerge. I started at 1.5 because I wanted a number that would hit a whole a couple times in our round but not make the “10” of them makes a whole number so obvious.
I did find that many of the same patterns emerged, however it definitely looked more intuitive for the students to look for patterns in the direction they had recorded.
Here are a few students who used the vertical recording…
You can see this student first noticed the 1.5 going up and down each column. She then noticed a diagonal pattern and could place the 9 where it would go had we continued.
This student started with thinking about them as whole number by multiplying them by 10. I love the last noticing because it makes such a beautiful connection to his first statement. When I asked him to clarify his thinking he did stipulate that you had to start at 3 for that to be true.
There is a beautiful statement in here that says she knew 0.3 is 10% of 3 because between each whole number there are ten 0.3’s. Lovely.
Here are some examples of horizontal, again, many of the same patterns…
This one was not so much focused on the patterns of numbers increasing or decreasing, but instead found that if you switched the whole number and tenths, the number would also be hit by a multiple of 0.3. Interesting to figure out why that works and when that doesn’t work. They left REALLY excited to keep working on this one. How much do I love the “I thought of this!” next to it!
I asked one student, who seemed content with his noticings before they shared as a table if he could think of any equations that matched the number patterns he saw while he waited.
I asked him where he saw the last one in the numbers and had him record it in Educreations: https://www.educreations.com/lesson/view/multiplication-decimals/31049585/
When the second class came in, I decided to switch up the number in each row to five (thanks Elham for that suggestion) to see if differences came out. Here was our board:
i definitely like the 5 in each row better than the 6, a lot more patterns emerged, quickly. It pretty much screamed patterns! We shared them all and I asked each table to pick one they wanted to explore deeper and figure out why it was happening.
This student said, “If you pick any number, go up and then over two the tenths digit will be one more than the starting number. It also works if you go down and then over two.” He explored that one here:
It was a wonderful first day back! My students and I really enjoyed the choral count (although they all spelled it coral:)! It was a very safe feeling knowing they were all saying it together, a bit different than the counting around the class.
-Kristin
MathMinds 
I think the 5 in a row and 10 in a row often makes for easier patterns to see two. But sometimes it depends on what you are counting by and what kind of multiples you end up on. So, for examples, counting by .25, you might put 2 or 4 in a row. Interesting to see how kids looked at when the whole numbers appear and how they paid attention to even-ness esp in a decimal context. Counting by three tenths is like counting by 3s. What if you did the count in fractions — is that more obvious? Thanks for jumping into this right away. Are there particular patterns you want them to try and justify? what would a good explanation sound like? Which patterns would they use to predict whether we would land on a particular number, like 12.7?
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Completely agree about the 5 and 10 in a row. My initial thought was that I didnt want them just doing doubling as they went down like when the 1.5 and 3 are right on top of each other. I wanted them to look past one step. Many get stuck on one number to another number pattern instead of over the course of the row or column. I thought 6 in a row would make for an interesting extension past one number to another. I def did like the 5 better though. I thought about fractions but I am really trying to get them to appreciate decimals a bit more. They love fractions and are very comfortable there but I feel they go there over reasoning about when decimals may be actually more convenient. So, I am trying to build their comfort in decimals as well. Fraction comfort took a while, as I am sure decimals will. I let them choose their pattern to justify today to get a variety bc 5 of my 6 tables had patterns that involved the number of jumps being made, which allowed this multiplication of decimal piece come out. A good explanation to me is one that could be used to predict future decimals based on the structure (number of decimals in each row/column) of the set of numbers. Of course, each student is at a different place, so a good explanation varies in complexity. Sometimes in other work we list our noticings and I choose the one to investigate, however today all felt within the same vain, so I let them choose. There were many patterns they could use to find 12.7 in my mind, but I hate to put words in their mouth. I think that would be a great question for tomorrow!
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It’s so fun to see you try this and your kids are so amazing by the way in what they noticed and got right into it. What I love about this simple counting work is how many questions can arise — I was just listing some that came into my mind before an appointment with a student of mine. I kind of like the recording of 6 like you tried. That’s the fun of all this, right!! Can’t wait to follow your adventures. Another student here…gotta run!
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So many questions is right! My Ss are so used to noticing/wondering bc it is such a part of our classroom routine that they are so amazing w/it! They find it as fun as I do! I find it fun to try some slight changes to see what happens! I knew number of jumps of .3 would come out of either (which was my ultimate goal for predicting) so why not see what else happens! Was also thinking on that diagonal pattern on the 6 in a row that it may be cool to ask if this pattern could make a square ending in a whole number at the bottom right.
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Love this, Kristin! And love reading your kids’ journals. Particularly since you’re thinking about multiplication and because you have a classroom community that would probably be comfortable with it, I think this count could lead itself to a good count around the circle. I’d be curious to hear what some of the estimates were before the count of the last number and whether they’d revise that half way through. I bet they’d have some greats strategies.
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It is amazing the thinking that goes on with a well thought out choral counting. Counting isn’t just for Primary students! You have made people think about the power of choral counting with decimals. I like to refer to this as “What really COUNTS!”
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Thanks Sandra!! That is SUCH a better title!! I am getting up with you before my next post for a title, brilliant! This student thinking really is what counts!!
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