The Mathematics of Core Sampling: Division of Fractions

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.

core 1

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:

core 3

core-4.jpg

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:

core 5

core-6.jpg

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.

core 7

core-8.jpg

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?

Final Thoughts

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.

Biking on the fractional number line . . .

The following post comes from Rafael Quintanilla, a Mathematics Instructional Coach for the Los Angeles Unified School District and a long-time teacher of number strings.

About eight years ago, I was teaching fifth grade and I wanted students to begin to see relationships between halves, fourths and eighths, be able to decompose fractions into unit fractions, and have an understanding of scaling when multiplying.  So I decided to write and try out some strings that focused on benchmark fractions as helpers and then go from there.  I used a number line as a model because that is what we tended to use as a fraction model because the students were able to connect it to real life applications.  I decided to use the context of a bicycle ride because students can relate to to riding bicycles and because they can picture a straight bicycle ride (I let them know I ride my bicycle on the riverbed next to my house), hence the number line.

Students are really engaged in this string because as the string develops, I tell a story of why we stopped at certain places. Some of the “look-fors” in the string is that students might  just give you numbers instead of saying the numbers represent miles on a bicycle ride.  Be careful because students will see the numeric patterns that exist but not make connections that these numbers represent fractional distances on the ride.

Multiplying Fractions by a Whole Number

Model: Number Line

Context: On weekends, my friend and I enjoy riding our bicycles. Our goal is 24 miles. But sometimes, we stop for breaks.

Draw number and teacher labels 0 miles on left side and 24 miles on right side

String:

1/2 of 24 (“tell the story, we stopped half way through the bike ride to get ice cream”)

(take one or two strategies, helper problem)

 1/4 of 24 (“Oh, I forgot, we also stopped one fourth of the way to drink some water”)

(take one or two strategies, helper problem)

3/4 of 24

(this is where the string begins to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

1/8 of 24 

(take one or two strategies, helper problem)

3/8 of 24

(this is where the string continues to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

7/8 of 24 

(this is where the string continues to develop, allow more time, ask students to think-pair-share and then model their thinking whole class)

5/8 of 24

(usually comes up during other problems or else you can use it as your last problem)

Two notes:

  1. Keep in mind that the goal of the string is to multiply fractions by a whole number. Therefore, the teacher puts the tick mark on the number and labels the fraction.  The students are computing the number of miles and developing strategies.
  2. The goal of the lessons is to allow students to discuss the relationship between the fraction and the whole. The goal is not to determine where the fraction is located on a line. That is why the teacher marks the location. Also do not worry about rushing into writing the expression or equation.

 

 

 

A count around for fractions

This February, I led a Number Strings Writers’ Retreat, as part of my role as staff developer at Math in the City.  One participant was 4th grade teacher Kathy Minas, a former colleague from PS 158.

Kathy wanted to design strings and other routines to help her students move past rote strategies (e.g., stacking) when subtracting fractions, but also to support them to think flexibly about the relationships involved in situations that involved fraction subtraction.

At the retreat, we began thinking about two central questions:

  • How could we use a visual component to help children during a fractions count around?
  • What context would support the students to reason about the quantities involved?

We chose the context of brownies as this is often something children can visualize. We used actual pieces of paper to represent the pans and pieces of brownies with the intention that kids would hold and move these pieces as the count around progressed. We hoped that this “manipulative in hand” would make the experience concrete, helpful and memorable.

But before she led the count around with kids, she and I acted it out with together. This helped us anticipate:

  • what kids in her class would experience
  • what part of the conversation she would record
  • what strategies her fourth graders would have
  • how she might support anyone who struggled or who needed a challenge

Below you will find Kathy’s notes from the count around.  We hope it’s helpful for you and your kids, too.

— Nicole Shield, Staff Developer, Math in the City

 

Materials:

  • Five wholes cut into fourths
  • An empty number line set up to 5 wholes
  • White board or document camera with paper for recording jumps on number line, equations and kids’ strategies

Start with 5 wholes cut into fourths displayed on the rug with the class sitting in a circle around it.

Introduce the context:

I want to tell you a story about my friend Sonya and I brought some materials for us to use to help us to visualize this situation. On Saturday, my friend Sonya baked 5 trays of brownies for her family, which included her husband and three kids. She cut each tray into fourths. Using the model here, can you tell how many fourths she had?  Turn and tell your partner.

Record what kids say — 5 pans is 20 pieces OR 5 = 20/4

After dinner on Saturday, Sonya brought out the trays of brownies. She ate ¼ of a tray. How many trays of brownies were left? How do you know? (4 ¾)

Record — 5 -¼ = 4 ¾

Then her husband Mike ate ¼ of a tray. So now how many trays of brownies were left? How do you know? Is there another way to think about this portion?

Record what kids say — 4 ¾ – ¼ = 18/4 or 9/2 or 4 ½

But be sure to push their thinking:

I thought we were talking about fourths. Where are the nine halves here? Who can show us in the model?

Mathematicians, are you claiming that that 4 ¾ – ¼ = 18/4 = 9/2 = 4 ½. Talk to your partner about whether or not you agree with this statement, and if you do, how would you convince those of us who are not yet convinced?

Count 2
4th graders preparing to convince each other. Photo taken with parent consent.

Bring the class back so that a student (or two) can try to use their model or other reasoning to convince others of this equivalence.

Well, Sonya’s family loves those brownies, so now each of Sonya’s three boys eats ¼ of a tray. Can you picture this, using our model?

You may want to invite a student act this out, using the shared model in the center of the circle.

So, what do we know now? What problem or problems did we just solve? Turn and talk.

4 ½ – ¼ = 4 ¼

4 ¼ – ¼ = 4

4 – ¼ = 3 ¾

OR maybe even….

4 ½ – 3/4 = 3 ¾

So, how many trays of brownies were left after dessert on Saturday?

Well, there is more to the story.  On Sunday, Sonya brought out the remaining trays of brownies. First, she ate ½ of a tray of brownies. So, now how many trays of brownies are left? (3 ¼ trays)

Record — 3 ¾ – 1/2 = 3 1/4

OR

3 ¾ – 1/4 = 3 1/2

3 1/2 – 1/4 = 3 1/4

So, now, using our model let’s make some predictions. Sonya just ate 1/2 of a tray for herself, right? I’m wondering: Are there enough trays of brownies left for the rest of her family to also each eat 1/2 of a tray? What do you think?  Will there be enough? Turn and talk.

Count 4
Kathy’s 4th graders during a turn and talk. Photo taken with parent consent.

Mathematicians, what do we think? How do we take away ½ of a tray from 3 ¼ trays of brownies?

Invite your students to use the model to act this situation out.  You might start by asking a student to just model what happens when Mike, Sonya’s husband eat his 1/2 a tray.

3 ¼ – ½ = 2 ¾ or 11/4

Together with your students, model the removal of ½ of a tray of brownies three times, one for each of the boys.

2 ¾ – ½ = 2 ¼

2 ¼ – ½ = 1 ¾

1 ¾ – ½ = 1 ¼

So, how many trays of brownies are left? How many fourths is that? (1 ¼ trays or  5/4 trays).

Here’s our last prediction. On Monday night after dinner the 5 members of Sonya’s family want to share the remaining trays of brownies equally. Is this possible? If so, how much of a tray of brownies would each person eat? Turn and talk.

Have a student or two act this out using the physical model, while you record on a number line, making note of the equations that correspond to each action in the model.

1 ¼ – ¼ = 1

1 – ¼ = ¾

¾ – ¼ = ½

½ – ¼ = ¼

¼ – ¼ = 0

So, now I’m thinking about this question: Over the course of these three days, how many total trays of brownies did each person in Sonya’s family eat? How do you know? (1 tray)

Two big ideas that emerged:

  1. When subtracting fractions, mathematicians may find it helpful to rename whole numbers into fractions with equivalent denominators.
  2. When subtracting mixed numbers, we may need to break apart wholes in order to make it work.
Screen Shot 2016-03-19 at 5.45.00 PM
Kathy’s notes (recorded on paper under the document camera) at the end of the count around

 

What the kids say…..

I’ve known Rachel Carr for many, many years.  She shepherded me through that exhilarating and exhausting first year of teaching and has remained a mentor and role model ever since.  This summer Rachel attended a summer institute at Math in the City, and we got to work together again — making sense of the landscape of learning for rational number.  I am continually struck by how a teacher with so much experience and insight still considers herself a learner.

Continue reading “What the kids say…..”

Division of fractions — The role of ratio

 

A 4th/5th grade teacher I am working with planned to introduce division of fractions and needed some ideas. I knew that a context would really help kids make sense of the operation so I nudged her towards the book Minilessons for Operations with Fractions, Decimals and Percents (Imm, Fosnot & Uttenbogaard, 2008). We found a good place to start and I offered to lead the first string.

After we discussed the word ratio, I set up a ratio table with # of cans of paint and portion (of wall painted).

Continue reading “Division of fractions — The role of ratio”

A Number String for Angle Measures — Before We Kick the Bucket

This post was co-created by Jesse Burkett, Ranona Bowers, and Adrian Sperduto — teachers in the Hazelwood School District and their colleague Cheryl Montgomery of the Parkway School District (both in St. Louis, MO). They were recently selected as participants in the Mathematicians in Residence program — a three year, three district project involving almost 100 teachers and eventually a summer math academy for approximately 200 students. Jesse, Ranona, Cheryl and Adrian just completed a two-week professional development academy, focused on numeracy routines in grades K – 5.  Jesse Burkett, a 4th grade teacher at Brown Elementary School, led the writing for his colleagues. Continue reading “A Number String for Angle Measures — Before We Kick the Bucket”

Searching for Friendly Numbers

Students who are working to become efficient with fractions must learn to seek out “friendly” numbers — a shifting target depending on the problem’s denominators. Once they recognize multiplication/division relationships, students can exploit the properties of multiplication to simplify computation. The following string encourages them to do just that.

string4

Continue reading “Searching for Friendly Numbers”

Fractions as Operators (Dot Arrays)

Here’s a collection of strings written by teacher participants at the Summer Institute at Math in the City (City College, NY).

When students share their strategies, you might ask, “How do you know?  How are you seeing it on the array?”  Then circle or shade what they saw.  Remember to open it up to other ways of seeing, “Did anyone think of it differently? Oh great. Ronald, what did you see?”  Then the second student’s strategy or envisioning is shown on a different array.  I like to print several copies of the array and have them ready to go up.  Otherwise, it takes too long to draw the dots each time. Continue reading “Fractions as Operators (Dot Arrays)”

Division – Whole number by fraction

During the spring of the 2012-2013 school year Kara Imm and I were working with my 5th graders to help them visualize what it meant to divide a whole number by a rational number. The students were very quick to invert the fraction and multiply. They loved saying, “Flip and multiply.” Mind you, I had never uttered those words in the classroom. However, this class was very used to string work and representing their work through models.

Continue reading “Division – Whole number by fraction”

Fractions as operators on money (Early fractions)

IMG_2820

Fractions as operators,

Money

What is ½ of $1.00?

What is 1/4 of $1.00?

What is 2/4 of $1.00?

What is 1/8 of $1.00?

What is 3/8  of $1.00?

What is 1/4 of $2.00?

Note: I modeled this on an open (double ) number line.

This is intended as an early fraction string, best done with kids as they are beginning to think about fractions, and already have exposure to the open number line.

Halving with early fractions

IMG_2821

Minilesson: Halving

(Rachel Lambert)

What is ½ of 1?

What is ½ of ½?

What is ½ of 1/4?

What is ½ of 1/8?

What is ½ of 2/3?

Notes: I modeled this on the open number line.

This is intended as an early number string when kids are beginning work on fractions.  Doubling and halving are great places to start to develop rational number sense.