Saturday, December 12, 2015

"Every fraction is a division problem!" -- 4th grader, Rowan

This week in 4th grade Room 118, we experienced the true value of math teachers collaborating across the country and around the globe! It led us to discover that every fraction is a division problem.


After I posted a fraction equal sharing lesson from Room 118 (12/5/15) I received a post from one of my twitter colleagues, Kristin Gray (@MathMinds) with a lesson idea to extend our fraction understanding.  Kristin had posed similar equal sharing problems to her students and then asked them to look for patterns in the problems and solutions...what did they notice?  What did they wonder? What did they discover? Fractions as division! (See Kristin's blog: Seeing the richness of Kristin's students' noticing, wondering and proofs, I was inspired to roll-out a similar investigation in Room 118.  What's follows is the account of our discoveries and connections, as well as some questions for you all.

Building upon the equal sharing work we had done the prior week, the students of Room 118 pursued the following 3 problems adapted from/inspired by Empson and Levi's book, Extending Children's Mathematics, Fractions and Decimals:
1) There are 4 burritos for 5 children to share.  If they each want to have the same amount, what fraction of the burrito would each child get?
2) Mrs. Cornn accidentally only bought 7 candy bars for her 8 nieces and nephews.  She wants them all to have the same share so how much will they each get?
3) We have 13 liters of juice for 10 children to share equally.  How much juice should each child get?
Figure 1A
Figure 1C
Figure 1B
During Math-by-Myself, many of the students recalled their earlier work with equal sharing problems and immediately began to recognize that there was a pattern...4 burritos for 5 children, 4/5; 7 candy bars for 8 children, 7/8; 13 liters for 10 children, 13/10.  (See Figures 1A, 1B, 1C)  One of these students walked right over to the Conjecture Wall and posted a conjecture about the pattern he was seeing. (See Figure 2)
Figure 2 Kaden's equal sharing conjecture
Another group of students started their work with the guess-and-check strategy...start by splitting the burritos in half, then thirds, then fourths and finally fifths.

While the students worked during Math-by-Myself, and also during pair-share, I collected evidence of students' levels of understanding so that I could scaffold various solution paths during our whole group discussion.  Today in whole group time I chose a student, who is more introverted, to share because she had definitely started to see patterns so I wanted to feature her work.  She described her work with Problem #1 as dividing up each of the four burritos into fifths because their are 5 children as seen in Figure 3.  
Figure 3 Dividing up burritos
We discussed why she would jump right to fifths and all of the students connected with her reasoning.  Before we moved onto Problem #2 I created a chart on the whiteboard and asked the group if they could fill in the chart based on the pattern of problem #1 and the work we had done previously.  The students could see the pattern unfolding as we created the chart in Figure 4.
Figure 3 Summary of Patterns in Equal Sharing Problems

When we got to the problem with 13 liters and 10 children we had a discussion about how this problem was different than the first two.  In this case there were two possible solutions, 1 whole and 3/10 and 13/10.  This was an a-ha moment for many students who had painstakingly divided all 13 liters into tenths.  "Of course!", they exclaimed, "Each person could have 1 whole liter and then the last 3 liters would be divided up among the 10 children."  (See Figure 4)  This realization is a true testament to the value of these type of discussions because I could have easily assumed they already understood that 13/10 was the same as one and three-tenths when only a handful truly did.

We extended our chart to include other possible problems so that we could test the pattern we were seeing.  Watching their faces as their synapses fire and their brains make these connections is always a sight to behold!

I felt is was now time to share Kaden's conjecture and to determine whether or not is was always/sometime/never true.  We have had dedicated conjecture-proof days in Room 118 on three occasions this year so the students are somewhat familiar with the process.  (I wish I had been blogging on those days!)  I was encouraged to pursue this process by my colleague, Mark Pettyjohn.  I was a little reluctant at first because of my own negative experiences with formal proofs in high school and college.  Mark said I should just take it to its fundamental form...I notice a pattern, I wonder if it is always, sometimes or never true...let's find out.  As always with Mark's advice, I am glad I pursued it.  So, today, with Kaden's conjecture, the students were comfortable with the process.  Some asked Kaden to clarify he conjecture, especially the last sentence, "Without doing it backwards."  Here is some background information on Kaden's conjecture to provide context for the sentence.  He originally believed that the numerator would always be the smallest number from the equal sharing problems.  I asked him about his solution to problem #3 where he had 13/10 as his solution.  He then saw the gap in his conjecture and added the sentence "Without doing it backwards," as a way to close the gap.  So, returning to the discussion, with that clarification students began to understand Kaden's thinking. Based upon our class discussion, another student, Andy, said he had created a modified conjecture based on Kaden's and he'd like the class to consider it. Here is his conjecture: "[The] Numerator is always the unit number (burritos) and the denominator is the people (children)."  A moment of silence settled over the class as students reflected on Andy's conjecture.  I could tell that some of the students were starting to grow weary, maybe they were confused, maybe their brains were full.  Consequently, I gave them 2 options: (1) Continue to work on refining Andy's or Kaden's conjectures in their Math Journals at their desks or (2) remain on the carpet with me to further understand the pattern we had noticed in the chart and how these conjectures evolved.  The students whose minds had started to drift came back to life and stayed with me on the carpet while the rest of the class found their next clean page in their journals and wrote their thoughts about the conjectures for the next ten minutes.

I was grateful for the time to share with the group on the carpet.  They had a strong understanding of the chart patterns but were less clear about Andy's conjecture.  It believe they had a clearer understanding as they talked and summarized what they knew and what they wondered.

We reconvened our whole class discussion and Andy was all excited.  He had an idea on how to make his conjecture more precise, in alignment with SMP6, Attend to Precision.  He said, "You know how in a multiplication problem you have factors that are multiplied together to make the product...what are the 'factors' called in a division problem?"  I asked if anyone knew and no one was quite sure.  (This was not surprising to me because the students have not spent much time with formal division up until 4th grade.)  I wrote a simple division problem on the whiteboard 12 divided by 4 equals 3 and defined the 12 as the dividend, the 4 as the divisor and the 3 as the quotient.  Another student, Sullivan, exclaimed, "Hang on a second, we need to write this down in our journals!"  (Don't you just love the enthusiasm of 4th graders!)  I then asked Andy what he thought would be the division equation for the burrito problem.  He said, "Four divided by five is 4/5, so the 4 is the dividend, the 5 is the divisor and the 4/5 is the quotient."  I will confess, at this very moment I had one of those moments when I can hardly contain my math teacher joy!!!  I asked Andy what made him connect this to division and he thought it was because we are dividing up the burrito.  He then asked to share his revised conjecture which is shown in Figure 5.  We then tested his conjecture with the numbers from our table and for that set this conjecture seems to always be true.
Figure 5 Andy's Revised Equal Sharing Conjecture

While Andy was discussing his revised conjecture, another student, Rowan had eagerly expressed an interest to add on to what Andy was saying but he patiently waited while Andy explained.  When Andy wrapped Rowan jumped up and said, "This confirms something I have been thinking about...every fraction is division problem!!!"  Inside my mind I, too, was jumping for joy but my external response to the class was, "Hmmm, that is interesting, Rowan...what do the rests of you think?"  The response was silent, reflecting minds and since the clock on the wall said it was almost time for recess we stopped to reflect in our Math Journals with the promise that we would return to this idea of Rowan's another day...soon!

I don't believe that most of the students have made the fraction as division connection yet but I am so pleased that it is out there as part of our ongoing conversation.

I am wondering where you, my colleagues near and far, would go next to strengthen this connection?  Also, what other connections did we we miss?  What are your thoughts on future Problems of the Day?

Also, what is your advice on creating a more rigorous conjecture-proof process, especially in a 4th grade setting?

I wanted to follow-up with you, Tracy Johnston Zager and Marily Burns, that this series of problems included more rectangular shapes that were not pre-divided and it did not seem to alter the students strategies much from the approached they used with the spherical, subdivided orange.  What connection might I have missed or could have expounded upon?

What's up next from Room 118?  Ms. Burns gave me an idea for a problem from Math Solutions's Math Reasoning Inventory that focuses on student understanding of equivalence.  (See: )  In an effort to explore all fraction models: area, set and linear, we are going to look into the linear model problem about who lives further from school, the student who lives 3/4 of a mile away or the one who lives 6/8 miles away?  Stay tuned!


  1. Once again - a wonderful lesson, and thanks for sharing it Nina. I love that someone said, hey we should write this in our journals when you named the parts of a division sum! It shows what a positive attitude you've encouraged in your students.

    We've been looking at fractions this week in our Year 4 (3rd Grade) classes this week. I find it hard to pace. Felt like I was pushing it too much on Friday (also 1/3 of them disappeared for 1/2 hour for choir practice, and then returned after the key discussion!) so I feel I need to somehow hand it back to them on Monday...

  2. Thank you, Simon!

    Fractions are definitely a challenge to pace. For me the key is to just make and take the time. I believe that we are establishing the foundation for our students fraction knowledge and it needs to be strong and secure for the rest of their lives. When I use to teach 7th and 8th grade math one of the biggest challenges was filling in the holes of the students fraction knowledge. Consequently, I will devote the majority of this year to fractions.

    Just a few minutes ago I was reviewing Michael Battista's book on fractions, Cognitive-Based Assessment and Teaching of Fractions. (You should check out his books if you haven't had a chance to yet! I gave an assessment this week and it appears the students have a solid understanding of the fraction concepts we've been exploring. However, when I look at Battista's book, which explains levels of understanding in detail, I notice how much we still need to explore. We will get there but it takes time.

  3. I'm curious about this lesson and the connection to division. How did Rowan get to his conjecture that every fraction is a division expression?? His conjecture from this lesson seems like the culmination of something he had been mulling over for awhile. What else had he been thinking about to make the connection? The likes of Carpenter and Battista have laid out some clear paths for how students progress to thinking about different concepts. What's that path for seeing fractions as division? I'm not sure, but I'd love to know more about these students' thinking and reasoning.

    On another note, it is fascinating to read about all of the mathematical goodness that came out of your classroom in this lesson inspired by Kristin Gray. This is truly a new era of collaboration.

    1. Thanks, Mark, for your question. We ran out of time that day for me to ask Rowan so I will circle back with him and I will also check his Math Journal.

      I am very grateful to Kristin Gray and the new era of collaboration. Thank you for encouraging me to pursue it!