—by Cheryl Leung, Golden Apple Fellow—
The aroma of freshly baking cookies filled the kitchen. The timer buzzed and I reached into the heat of the oven to pull out a pan of giant chocolate chip cookies. As I placed them on a rack on the counter to cool, I played with patterns – a row of two cookies, a row of three cookies, and another row of two – and I smiled. Like all things, these cookies make me think about math. Earlier in the day, these cookies were my answer to the question “What is division? What does it mean to divide?”.
Every year, I teach my students to model mixed number division. This is a really difficult thing for most people (including adults) to do. It is hard, because, oftentimes people have only a partial understanding of what division is. Here is some of the thinking that my students shared.
In looking at their responses, it is clear that many of them see division as breaking something into a number of equal-sized groups. While that is a good start, it is only part of the story. That is where the cookies come in. It is a tale of two stories, because stories help bring ideas to life.
I made a batch of 2 dozen cookies . I decided to bring them to my student aides as a thank you. I have 4 student aides. How many cookies will each student aide get?
This story echoes the student thinking on division. It is the story of dividing something into an equal number of groups and figuring out how big each group is.
It is an important story, but there is another important story as well.
I made a batch of 2 dozen cookies. I decided to make bags of 6 cookies that I could use as a thank you for my student aides. How many bags can I make?
This is the forgotten story of division. It is the story of dividing something into a certain size group and figuring out how many groups of that size you can make.
Together, these two stories explain what division is. In order to effectively model division, students must see both stories. It is really, really hard to model something that you don’t understand.
In introducing division models, I utilize both an array model (brownie pan) and a ribbon model. I start with problems in which students know the size of a group and have to find the number of groups. In this problem from Connected Math, students are told that they have nine bars of cheese and the amount of cheese needed to make a single pizza. They then need to determine how many pizzas they can make.
Then, I introduce problems in which students know the number of groups and need to find the group size. In this problem from Connected Math, students are told that they have a certain amount of peanuts that will be shared equally among a certain number of students.
Finally, students have to generalize and determine what kind of a problem it is and model accordingly.
My favorite thing about this lesson yesterday was watching the girl who has been struggling a little bit. As my glance danced between her work in Desmos and her face in the Google Meet, I could see that magical look of celebration as she was working on a problem and seeing it fall into place. The model was helping her make sense of the math. It was a “cookie worthy” moment, but the cookies will have to wait for a different time.
Back to the cookies coming out of the oven. They also made me smile because they are from a recipe that one of my student aides gave me last year. I had brought him and my other student aides baked goods from time to time as a thank you for all that they do to help me. He decided to bring me cookies that he made one day. He is a pretty serious foodie and his cookies were the kind that you want to savor. In addition to the cookies, he gave me his recipe. Every time that I make them, I think of that amazing kid and it makes me smile.