Exploring fraction operations with Choral Counts

I think sometimes it’s easy to see Choral Counts for their glitter, and miss their substance. People quickly fall in love with them because they’re fun, engaging, interesting, and they get kids talking about math. And these are great reasons to love and choral counts!  Beyond all this fun, though, choral counts (and the representations they produce) can support students to investigate really critical concepts – even in the older grades.

Yesterday I worked with some 5th grade teachers to plan some choral counts they can use in their current unit, which is all about fraction multiplication and division – specifically, division of whole numbers by fractions, and vice versa. Can choral counts support students in this unit? Absolutely.

Check out this count we went through yesterday. We counted up by 1/3, starting at 1/3, using rows 4 across. I specifically chose to use rows of 4 in this count to make it “offset” in a way that would create some neat patterns. We found lots! What patterns do you see in this count?

Here are some we found:

  • The down-left diagonals increase by 1.
  • Three hops ‘down’ the column is an increase of 4 (this is especially apparent for the whole numbers, so that’s where I marked them.)

Each hop ‘down’ a column is an increase of 1 1/3.

  • There are some “missing” whole numbers in each column: for instance, in the third column, there’s no 4. In the fourth column, there’s no 7.

By drawing a little box where all the missing numbers are, another interesting pattern emerges…

Ok ok ok, we could do this all day. But what about exploring fraction multiplication and division with this count? 

Consider posing the problem 8 ÷ 1/3. How would you solve this problem mentally? How might 5th graders?

Now, try using the finished count to solve this problem.




(Think about it on your own before you scroll down.)




One way of thinking about 8 ÷ 1/3 is, “how many 1/3 counts does it take to get to 8?” So we can get the solution by counting our ‘hops’ in the count. Because we didn’t start at 0, we count the numbers, not the hops.

Something that struck us in our PD meeting yesterday was that this is a nice way for students to confront their (totally reasonable) preconception that dividing always ‘makes something smaller.’ But how can we hammer that home?

Consider adding more problems in a string – building larger divisors. How could you use the count to figure out
8 ÷ 2/3? Students first have to figure out what 2/3 looks like. Can you build 2/3 by grouping 1/3’s? Sure! So how would that help us solve the problem? Count the hops again – only this time, we’re counting 2 hops at a time.

Something we’d want students to really be firm on: 2/3 is larger than 1/3. It’s a larger divisor.

So, let’s check out one last problem:  8 ÷ 4/3. [First: we want students to make a mental note. 4/3 is larger than 1/3 and 2/3.] Woah woah woah – dividing by an improper fraction?? Why not? Again, students need to first figure out where 4/3 appears in the count: can you build 4/3 by grouping 1/3’s? sure!  So using the same process we did above, we build a larger divisor and count those hops.

So now, let’s check out this string of problems and solutions that we’ve created. What might students notice? What big ideas might this string help them to make sense of? What questions might you as a teacher pose to students to get them thinking about those big ideas?














So, in sum, here are some big concepts you might explore in a choral count like this one:

  • Fractions can be decomposed into unit fractions – and unit fractions can be grouped together into larger chunks – even into chunks larger than 1! This is actually a 4th grade standard, but it’s very likely that 5th grade students need more opportunities to see this concept in action.
  • Dividing something relatively big by something smaller means fitting small things into something big. So we shouldn’t automatically expect the quotient to be a smaller number than the dividend. This is essentially applying a 3rd-grade understanding of the meaning of division to fractions – and thereby arriving at a key 5th grade standard.
  • The larger the divisor is, the bigger the thing is we’re trying to ‘fit’  into the dividend. A related idea is that the closer the divisor is to 1, the closer the quotient will be to the dividend.

Have you used choral counts in your classroom to explore and model similar concepts? Tell me about it in the comments – I’d love to hear!

Fraction Kits – beginnings

So far in summer school, we’ve been working on sociomathematical goals: getting students accustomed to routines and procedures for sharing their thinking in pairs, in written work, and in whole-group dicussions.  Mathematically, the activities and lessons we’ve used to develop these goals have been focused on multiplication and the relationship between skip counting, grouping, and array models of multiplication. This has helped us identify students who need extra support (since multiplication isn’t ‘new’ content for 3rd grade) while also being a slightly more approachable area to start number talks with.

Last week in our classroom, we began work on the content standards that gave our 3rd graders the most trouble on the end-of-year assessments administered by the state and by the school: fractions. Specifically, these students need additional support in understanding what a fraction represents, how fractions can be represented on linear models, and how fractions can be different and yet equivalent. To start this work last week, we began by creating Fraction Kits based on the insight and advice of Marilyn Burns, whose book has been an excellent tool for lesson planning and mapping the mathematical terrain we need to help students explore this summer.

Monday and Tuesday we spent time counting collections, and observing how students were grouping objects.

Wednesday we began math class with a mental math number talk, solving and discussing strategies for 6×7. I was glad to see that students were willing to persist even if they didn’t “just know it” from memory – they were willing to think and find a means of figuring out the problem.

After our number talk, students returned to their seats where each received a set of pre-cut strips of construction paper and a large envelope cut from similar construction paper. They wrote their names on the envelopes and tucked them inside their desks.

Next we went through the process of folding our strips. For each strip, we:

-folded (I first modeled how to fold them)
-predicted how many sections there would be before opening to see
-decided as a group how we might then label each section
-waited for each student to fold, unfold, label each section, and label the back of each section with initials.

Once every student had their fraction pieces folded, labeled, and affixed with initials, we passed out scissors and got them cutting.

Why do, We do, You do?

I want to note that ordinarily, I steer away from “I do, we do, you do” type teaching. However, for creating fraction kits, I modeled first, together we decided how we might name/label, and then asked students to do it on their own. The reason I thought modeling and duplication was fitting in this instance is that our goal was for students to create tools that they could then use to explore fraction relationships. If their kits weren’t created correctly (and some weren’t: a couple students mis-folded and needed help to tape or re-cut later), they might develop some sneaky misconceptions about how much bigger 1/4 was than 1/8, etc.

So why not pre-cut the strips?

Giving the students a chance to fold and cut did elicit some students’ initial ideas about what those weird hyroglyphs called fractions might mean: many students were catching on to the pattern that “8 pieces mean each is called 1/8” and so on. Folding their own strips was also a useful opportunity for them to consider what happens when you take a half and fold it in half. For instance, when we folded the strip into 1/4ths, many students predicted that we would unfold the strip to find 6, 8, or 10 segments, rather than 4. Similar guesses of 10, 15, and 20 came out when we folded the strip into 1/8ths by folding it in  half, half again, and half again.

Wrapping up

Because we opened with a number talk, we only had time to get the strips cut, labeled, and put away on Wednesday. But we did close by asking students to see if they saw any “matches” between strips of different size. This got a few initial ideas out on the table (or in our case, the smart board): two 1/4th sized pieces was the same as one 1/2. Similarly, two 1/8 sized pieces was the same size as 1/4.

The Apple Task: Enactment artifacts from a 5th grade classroom

On 3/3/17, I was able to work with an outstanding group of 5th graders at a local elementary school on one of my favorite 3-act tasks by Graham Fletcher – The Apple.

The Apple gets students thinking about division as measurement, and if left to solve the problem in a way that makes sense to them, they tend to generate a range of strategies that can be really useful stuff for making connections in a pre-reveal discussion about their strategies. In particular, this task is useful for making connections between:

  • repeated addition (adding one block at a time until we ‘arrive’ at the apple’s weight)
  • repeated subtraction (taking off a block’s-worth of weight from the apple until we ‘arrive’ at zero)
  • multiplication (often in the form of guess and check after students grow weary of repeated addition), and
  • fraction division using the common denominator method (get a common denominator and divide the numerators).

We started by playing the video for students and asking them what they noticed and wondered. As they shared, we recorded their noticings and wonderings on chart paper:

As a class, we decided on the starred question as the one to solve for the day. Once we’d come to consensus (using words and gestures) for what it would mean to balance the scale,  we asked students to estimate the number of blocks it would take. We recorded their estimates.


We then asked students what information they would need to solve this problem – and why they would need it. Most students suggested fairly quickly that we would need to know the weight of the blocks and the weight of the apple – but I asked several students to argue for why we needed both and not just one. This took some time, but by the end it seemed most students understood why both weights were important, and what significance they had mathematically and contextually.

Then, we gave them the information. (I wrote the info down in the wrong ‘spots’ on the “what do we need to know” chart at first, so I had to rewrite.)

Once we gave them this information, we gave students about 2-3 minutes to begin a strategy on their own piece of paper – enough time to make a plan, but not enough time to follow it all the way through.

After a few minutes passed, we asked them to talk to the other students at their tables and discuss their strategies, and come to a decision about how they would use one or more of the strategies at their tables to solve the problem. They were also tasked with representing one strategy on a group white board. Here’s what those white boards looked like by the end:






Multiplication strategy: what number multiplied by 3/8 gets us 5 and 2/8?


These students started by repeatedly adding 3/8 until they grew weary of it, and decided to start multiplying. They tried 1, 2, 3, 5, 10, 13, and then finally did 13 blocks plus 1 more block.


Repeated addition of 4 blocks at a time.

I LOVED this strategy. This group was led by two students in particular whose really wanted to find a combination of blocks that would add up to an easy ‘split.’ So they settled on 4 blocks, since 4 groups of 3/8 makes 12/8, which would only need 1/2 an ounce to get to the next whole. They knew, then, that two groups of 4 blocks would ‘make up’ the next whole.

Once they started down this path, however, they found themselves in improper fraction land quite quickly, which it turned out was not a land they had adventured in many times before. Out of the 4 students in the group, 3 of them were quite insistent that 4 and 16/8 oz (which they arrived at by repeatedly adding 4 blocks 4 times – so, 16 blocks) was not yet enough weight to equal the apple. They had some difficulty moving past this moment, and ran out of time to come to consensus as a group about what should happen next – though they agreed that it would likely entail adding single blocks to finally get to 5 1/4 oz, since most of them were sure that 4 more would be too many.


Repeated addition, using a choral count representation

This was a strategy we were really hoping would turn up, since the students had just done a choral count by 1/8 in the moments preceding the 3-act task. Interestingly, this strategy did not represent the first strategy all of the students in the group had used; several had used a division strategy, but because they could not figure out how to represent or explain how that strategy worked (which was what we insisted they be able to do with any strategy they put on their board), they resorted to using a repeated addition strategy. The challenging thing in this group’s case was figuring out in all of this representation where the answer was.

In addition to these three group boards, there was one last group who had worked hard to find solution strategies independently, but who had run out of time to get much onto their board before we discussed strategies:

The students had about 20 minutes in their groups to produce the representations above, and then we took about 10 minutes to look across the first three boards above, first looking at the choral count strategy, then the repeated addition of 4 blocks, then finally the multiplication strategy. The questions I asked of each group were –

  • where is the apple represented in this group’s strategy?
  • How did they know where to stop?
  • What were they looking to have happen by [adding 1/8 at a time…adding 4 blocks at a time…multiplying numbers by 3/8]
  • How did they know they had done enough?
  • Based on all of this work, what is the group’s answer to the question?

Though I’m not confident that all of the students came to consensus about whether or why multiplication was more efficient than repeated addition, they all agreed that in this case, anyway, it produced the same result. The group who was adding ‘batches’ of 4 blocks at a time seemed to understand by the end that their method was functioning pretty well, but that it might require more adjustment because of undershooting or overshooting the perfect amount – a problem the choral count group didn’t have, but that the multiplication group did (they found that 13 was close, so they adjusted from there).

Afterward, we gave them exit tickets to see how they would solve a related problem – and to see how they were making sense of the fraction 3/8 as it relates to 1. Here are a few examples of student work :


All in all, this task was a great way to feel out students’ thinking about the relationships between addition, subtraction, multiplication, and division, and whether those relationships behave predictably with fractions. It was also a great way to peek into how students are marshaling procedures like finding a common denominator as steps to solving a problem, rather than as a procedure for it’s own sake. It’s clear that we have more work to do to build common understanding about some of these ideas, but I loved the opportunity we got to ask students to completely fly blind into material they’ve never ventured into before, and see what sense they made of it.

Counting up by 1/4 from 1/4

Note an additional representation on the right side (the open number line) where we explored what the “hop” was from 1/4 to 1 and 3/4.