About the Co-Authors
Keely Zaientz and Corey Levin teach 6th and 8th grade math in an integrated co-teaching classroom at Yorkville East Middle School. They have a progressive, constructivist classroom, centered around students developing and identifying their own strategies to approach problem solving. They are huge fans of number strings and spend a lot of their time trying to get better at teaching.
Number strings have become an essential part of our classroom culture. We frequently use them to launch a new unit of study or to reinforce a topic that students have started exploring, but have misconceptions that need clarification.
Our students frequently struggle with the meaning of fraction division. This is a topic that makes a great deal of sense intuitively; however, when students need to identify strategies to support their intuition it seems to violate some unwritten rule about fractions that just makes it so confusing. This fraction string was developed to utilize models to support an understanding of what fraction division means before introducing students to the notation of fraction division.
At the start of each string, we gather the students close to the board and have them bring their notebooks and pencils. This allows them to stop and jot their thinking as situations are discussed.
For this string, I (Keely) share that my brother is a geologist and we are going to explore a tool that geologists use called a corer. We show them some pictures of the coring process and ask students to identify their noticings.
We usually look for noticings like, “Part of the corer is in the water and part is out of the water” or “It is taking a measurement of some sort.” These observations get us closer to being able to mathematize the work of the geologist.
We often provide some background information as well, like the idea that it is so cool that geologists can determine how the environment has changed over many hundreds of years simply by taking core samples. A core sample is collected with a corer that you put into the sediment and it pulls up buried layers of wet mud in the order of it settling. The sampling that is going on is happening in a relatively shallow part of the river, near the banks. Geologists use a corer to bring up buried mud in an effort to study it. Once we feel as if our students have an initial understanding of this context, we are ready to introduce some values for them to model and eventually make sense of.
The first situation goes like this …
At the core sampling site, I noticed that there was a geologist with a corer that was half in the water and half out of the water. The part that I could see in the air was 2 feet long. How long was the corer?
Students have some time (about 1 minute) to discuss the situation with their peers and are asked to model their thinking.
We share how solutions were identified and we show one model of the situation on the board. Frequently, one student draws a corer partially in the water, and out of the water with 2 feet labeled on both parts. If a student doesn’t volunteer this information, we often try to solicit it from students or co-construct a sketch with our students, so that we all have one clear model. This model stays on the board and becomes a scaffold to all learners.
Here are some pictures from student notebooks and the noticings that we have jotted on the board:
In the second situation …
I got to the core sampling site, I noticed that there was a corer that was ⅓ in the air, 1/3 in the water, and the rest was in the mud. I was told that the part in the mud was 2 1/3 ft long. How long was the corer?
At this point, we usually allow students to take one minute to jot their own thinking followed by 2 minutes with a partner to develop a model that will tell the story. If students finish early, we are ready to challenge them to model their thinking with an equation.
Again, we have a group discussion about the models that were developed and any ideas they have generated about developing an equation. It is typical that students identify 2 feet by 3 parts of the corer will give you 6 feet, but will be unable to use 1/3 in their model. We allow our students to develop hypotheses related to how they deal with the extra foot and how they “split up” the numbers that they are interacting with each other.
The student notebooks often look like this:
In the final situation …
The best core sample was about to happen. I could tell by the corer they were using. The corer had 3 ft out of the water. I was told that 1/2 of the corer was in the water and ⅓ of the corer was in the mud. How long was the pole?
Again, we use the routine of having students think about it for one to two minutes and then talk with a partner. With some classes this takes 5 minutes and with others it can take 10 minutes or more. Again, if some of your class finishes early, challenge them by developing an equation to represent the problem and forcing them to prove why 3 divided by 1/6 is 18 feet.
The discussion for this situation usually focuses on dividing the corer into sixths and why 3 divided by 1/6 is the same as 3 times 6.
It is not unusual for students to confused the idea of 1/2 or 1/3 of the pole with half or a third of a foot. Typically, though, a model of the situation (that resembles a vertical open-double-number line) can help rectify this confusion. Seeing the proportional relationship between the fractional amounts and actual amounts sets our students up well for the upcoming work of proportional reasoning in 6th grade. In this string without telling our students that we are dividing, we offer a situation in which division is about rate and ratio (partitive or fair sharing division). We are nudging our students to associate two quantities (with two different units) in the form of a ratio: number of feet with portion of the entire corer. They will come to trust that this is division, even though many will not recognize it as fraction for awhile.
When we have a group that is really excited about the problem and ready for a challenge, we offer them this extension, which we now offer to you, too:
In this situation, the corer had twice as much length in the water as in the air. There was three times as much length in the mud as in the air. The corer was a total of 14 feet long. How long was the section in the air?
We are convinced, after years of using number strings in our practice as individuals, and more recently, as a teaching team, that they are a powerful tool that give students a real world context in which to explore math. Making a model of the situation is a norm in our classroom and our students have come to expect it. They often see problems without numbers and are asked to first “model the situation,” a way of encouraging sense-making before computing/solving. The practice of number strings also helps us develop a community where dialogue about problem solving is at the center. Wherever possible we use a context, because we have seen the power of reasoning about mathematics in a realistic and/or believable context. So, we hope this string inspires you to a) try it with your students b) write a version of it that your students would love even more and/or c) leave a comment with your ideas for us here.