Yvette Lehman, Teacher Consultant, GECDSB
New conjectures about division
From April 3rd through 5th, I had the opportunity to attend the NCTM annual conference in San Diego.
On Saturday morning, I noticed a tweet from my math education hero, Cathy Fosnot.
So at 10am, I made my way to her booth to participate in her mini-lesson. And true to Cathy’s style, she had a crowd of people wrapped up in her dynamic and carefully crafted string. The audience worked their way through a series of division problems and we, a group of educators, made new conjectures about division through conversation with our elbow partners.
Cathy made us feel competent and capable as learners of mathematics. She modelled on how to talk to students in a way that makes them authors of their own ideas and champions of their own clever strategies.
We can paint ½ of a room with ¾ of a can of paint
Being someone who has committed a lot of time to understanding division, I was amazed at the conclusions that were drawn from Cathy's presentation. We looked at the implications of doubling the dividend, doubling the divisor and doubling both.
However, that was just the warm up....
Cathy had intentionally scaffolded these generalizations so that we could apply them when further asked to divide fractions.
The scenario that Cathy presented was “we can paint ½ of a room with ¾ of a can of paint”.
She wrote the two quantities in the ratio table and labelled the columns.
Then she asked the question “is this division?”
Is this division
As my elbow partner and I engaged in conversation, we agreed that it was in fact division, but we differed in our articulation of the type of division. It was not until recently that I even knew that there were two types of division. It was only revealed to me last year through the modelling of student thinking on the number line.
My partner felt that the question was asking, how many ¾ of a can of paint are in a ½ room. This explanation is an example of quotative division, where you are dividing by a rate (taking the whole and dividing by how many per 1).
I was asking the question, if ¾ of a can of paint is used for a ½ of a room, how much paint is needed for one whole room. This is partitive division. The cans of paint are partitioned in a ½ part, and in partitive division, we are always asking how much per one (which reveals the rate).
This conversation created an opportunity for Cathy to clarify the difference between partitive and quotative division. Cathy used the example, if I have 8 cookies, I can divide the cookie into 4 bags, and I will have 2 cookies per bag (reveal how much per 1 part), which is partitive. Alternatively, she said, if I have 8 cookies, I can divide them by 4 cookies per bag (the rate), and I will have two bags. Both of these scenarios can be represented as 8 ÷ 4.
Once we had established that we were in fact dividing our ¾ cans of paint by a ½ part of a room, we looked back to our earlier conjectures. Cathy asked how we could double ½ in the ratio table to find the equivalent paint for one whole room. We considered our earlier string and remembered that to double we could double the dividend, so ½ became 2⁄2. We knew that if we want the quotient to remain the same, we should double the paint as well. To double ¾, we used the same approach, we doubled the dividend and we had 6⁄4 cans of paint. Using our understanding of relationships and the ratio table as a model, we were very confidently dividing ¾ by ½ without any reliance on a standard procedure.
How do students represent partitive and quotative division
Inspired yet again by the power of Cathy’s number strings, I took the opportunity to speak with her after. I had some questions about the implication of the types of the division on students’ conceptual understanding. Cathy graciously handed me over her marker, and allowed me to model for her (the queen of models herself) the ways that students in my grade 6 class represent quotative and partitive division on the number line.
There you have it
My time with Cathy was both inspiring and validating. I was already a fan of her work, but this opportunity challenged me to discover even more connections as revealed through her string and our conversation that will continue to promote my number fluency.