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


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


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


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.

Life beyond the algorithm: Division of decimals

About the author: Kit Golan

Kit is an MfA Master Teacher teaching 6th and 7th grade math in a NYC public middle school. He is dedicated to crafting experiences for his students that create cognitive dissonance to develop students’ mathematical mindsets. He meets students where they are, and challenges them to grow their brain and delve deeper into mathematical understanding. He is constantly reflecting on his own practice: sharing those reflections in his blog and tweeting at @MrKitMath

Challenging my Algorithm-Loving Students to Think

Recently I designed a sequence of strings to support my 7th graders to reason about division of rational numbers. I wanted to move away from the “algorithm only/always” approach I had seen and help my students build a bank of smart strategies. Ultimately I am hoping that my students are flexible thinkers with deep number sense, so this set of strings was designed to explicitly invite them to try new, different strategies based on the relationships of the numbers in the problems.

In our investigation of division strategies, I launched our first number string by telling students to look for relationships they could use to make division easier. Our goal for the week was to think about when long division was necessary and when there were more efficient or better strategies that could be applied. Our first string was designed to have students notice the constant ratio — when both the dividend and divisor are multiplied or divided by the same constant — also known as scaling up or down down division problems.

32 ÷ 4

320 ÷ 40

3200 ÷ 400

3.2 ÷ .4

5.6 ÷ .8

Students noticed quickly that we could scale the problem up and down to make friendlier numbers — and that the quotient stayed the same. They described the division as a fraction, and related the scaling to simplifying fractions. They emphasized that it had to stay equivalent, but we could be flexible in changing the numbers. They were able to apply this strategy to the new problem without a helper.

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Later in the week, I returned to the work and said the following, “This week, we’ve been working on division strategies, and considering what’s the best, most efficient way to solve a problem. Today, we are going to do a number string with a bunch of division problems. Our focus is not going to be on speed, because it’s not a race. Instead we are going to try to find the most efficient or easiest strategy to use. Our goal is to think like mathematicians — be strategic and efficient. Consider the numbers for each problems before you choose a strategy — and be ready to explain how your strategy makes the problem easier to do mentally.”

For my first period class I planned this string

13.2 ÷ 1.1

3.6 ÷ 1.8

7.2 ÷ 4.5

245 ÷ 3.5

32 ÷ .25

3600 ÷ 1.4

(108 ÷ 2.4) — planned but didn’t get to it

We did all of the problems, except the last one. Because of conversations we had earlier in the week, my students trusted that they could scale up or down a division problem to make it friendlier. This meant 13.2 ÷ 1.1 became 132 ÷ 11. My students knew that those two problems were equivalent. Though they were good at scaling by any power of ten, they did not take up the idea of scaling by other factors. Many of them got stuck on 7.2 ÷ 4.5 for example. They initially thought to scale the problem to 72 ÷ 45, but written as a division problem seemed not to help them. When it was written as a fraction, however [72/45] students knew they could rename it as 8/5 and later it became 1.6.

Screen Shot 2018-01-15 at 1.42.55 AMScreen Shot 2018-01-15 at 1.43.16 AM

Next I posed the problem 245 ÷ 3.5 and students were not sure what to do. Interesting things happened:

  • A student scaled the problem to 2450 ÷ 35, then changed the problem to 2450 ÷ 70 (by doubling the divisor) to make it easier for him to think about. Next he used partial quotients to build his way up to the quotient — essentially 2100 ÷ 70 = 3 and 350 ÷ 35 = 10. Once he had the partial quotients for 2100 ÷ 70 and 350 ÷ 35, he multiplied 2 x 10 x 3 to get 60 35s in 2100 and 10 35s in 350, which he added together to get 70. I was struck by the power of his working memory to hold all of these parts together and knew that recording his strategy as he spoke it would help him, and all of my other students make sense of his thinking.
  • Another student scaled the problem inconsistently — 245 ÷ 5 became 2450 (scaled by 10) ÷ 350 (scaled by 100). Later the same student adjusted the problem by a factor of 10 to accommodate for the original move. I was fascinated by this strategy — adjusting the problem to make a non-equivalent, but friendly problem, and then adjusting it back to make it equivalent again.

The next problem — 32 ÷ .25 was surprisingly easy for the students to solve. I think they recognized .25 as 1/4 of a whole, whereas they do not think of 3.5 as 1/2 of 7. Many may have thought about money as well — envisioning the .25 as a literal quarter, four of which are equivalent to $1 and thinking about how many quarters in $32.

For the last problem — 3500 ÷ 1.4 — I was surprised by the number of my students who simplified the problem by a factor of 7 — 500 ÷ .2 and then “just knew” it would be 2500.

Finally, I borrowed a practice from the Contemplate then Calculate [] routine and asked students to reflect on their own thinking [meta-cognition]. I reminded them of our goal of thinking like mathematicians and finding new strategies for new problems. I allowed them time to choose a prompt and write a response on an index card that I collected.

Screen Shot 2018-01-15 at 1.48.25 AMAfter reading through their exit tickets it was clear that students were in many different places, with respect to division of decimals:

  • Some mentioned using “common factors” as a helpful strategy
  • Others mentioned noticing “common multiples” as a helpful strategy
  • Some mentioned “scaling up or down” to make the numbers friendlier/easier
  • Many noticed patterns but not all could describe them or say what was helpful about them
  • Several hadn’t yet developed the language to describe the mathematics and wrote in vague terms — “having strategies that worked quickly”

For the next day, I planned an “entry slip” where students were asked to solve another decimal division problem using mental math and then ask them to record their strategies on an index card. I thought about using 108 ÷ 2.4, but initially worried it required too much scaling:

108 ÷ 2.4 = 1080 ÷ 24 = 540 ÷ 12 = 270 ÷ 6 = (240 ÷ 6) + (30 ÷ 6) = 40 + 5 = 45

Ultimately, I decide to try it out — not as a string, but as independent work, where students were asked to explain their thinking in words as well as in numbers. I asked student to “think about the strategies from this week’s number strings and use them to solve today’s problem.”

When I analyzed their entry slips, their work fell into a few big categories:

Scaling the problem up and down until it feels friendly (Hiro)

Screen Shot 2018-01-15 at 2.03.12 AM

Multiplying the divisor (2.4) by 10 to get rid of the decimal, then adjusting (Wendy)

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Getting rid of the decimal (Ming)

Screen Shot 2018-01-15 at 2.03.33 AMLooking inside the numbers for common factors (12), then scaling up by 10 (Dante)

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Looking for common factors in the numbers (12), then solving (Franklin)

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Scaling up the divisor to get rid of the decimal, then “making it equal” (Janice)

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Where are we now?

Initially, I saw many of my students struggle to solve division of decimals problems on a pre-assessment. So I was fascinated with how many of them were able to do our entry slip problem using strategies that had emerged in our strings. It’s clear from the exit tickets, too, that most of the students were able to use the strategies, while a handful of them were resistant to leaving long division behind. Sure, there were some students who made some calculation errors, but this was true of those who used long division as well as those who scaled the problem to a friendlier place.

Where am I now?

I am also thinking about how differently I lead number strings — from just a year ago. I know from other routines (Contemplate then Calculate) that very focused turn-and-talks at specific points in the routine is really important. I also watched Kara Imm (Math in the City) do this with middle school students at Lyons Community School (Brooklyn) this fall — a way to give all kids an opportunity not just to think, but also to talk. My students had better stamina this year, and they were more interested in listening when I asked them to put their pens, calculators and notebooks away. Engagement was better both because of the structure of the routine as well as the way we designed the strings to build from one to the next.

By crafting an opportunity for students to see the efficiency of other strategies over the standard algorithm, I encouraged my students to move beyond the algorithm as a standard default. Now, instead of mindlessly attacking a problem with a brute force strategy such as long division, my students are beginning to think flexibly about other possible strategies. This is evident from the number of students whose exit tickets show no signs of long division!

I’ve found a few things are key in delivering a successful number string. First, the sequence of the problems needs to guide students towards specific strategies and expand their tool box one piece at a time, without narrowing their focus too much on one tool, such as when I accidentally blinded my students by providing them with too many scaling by 10 and not enough “obvious” scaling by other factors, such as 2, 3, 4, 5, or even 12! Second, though the number string is a whole-class activity, it can and should be broken up into chunks of partner talk where students discuss their strategies in their partnerships and then discuss the strategies that are shared out. Finally, the reflection component at the end of class is critical for ensuring that students learn strategies to solve future problems and not just one solution for one problem.

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


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 “juicy” dilemma

This number string is inspired by an investigation (Comparing and Scaling, Investigation 1.2) from the Connected Math Project curriculum for 7th graders.  It can be used as a way to extend and solidify ideas that will develop as kids investigate, or it could be used to launch the task. In the investigation, students are asked to compare the following orange juice recipes, according to their “oranginess.” Continue reading “A “juicy” dilemma”

So, what’s the story?

This number string attempts to do something a little different it begins with the model (instead of the problem), and uses the model as an anchor to make sense of the context.

This means inviting students to “tell the story” to explain what is happening in the model. Then the story (and the model) expand, while the ratio stays the same. It is modified from ACE problems from the unit Comparing and Scaling (Connected Math Project, grade 7), and is intended to serve as a template for how existing curriculum can be used to design new flexible, interesting numeracy routines for students. Continue reading “So, what’s the story?”

New percentage number strings

We recently saw the problem above on a New York CCSS-M 6th grade benchmark test. A few teachers I was working with argued about what the question required.  Some teachers thought this called for converting a percent into a decimal and multiplying.  Another teacher reasoned persuasively, “I don’t want kids to do any calculations here.  None. Without a calculator they would be tedious and time consuming. And I also don’t want them setting up some equation like .12x = 3.84.  Unless they are really fluent in algebra and division of decimals, and there is a lot that can go wrong there.”

Continue reading “New percentage number strings”

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”

Multiplying fractions: Why context matters

Our fifth grade team was trying to encourage students to use a visual model to represent their thinking when they multiplied fractions. So many students were so fast — multiplying the numerators, then multiplying the denominators — but had no context and demonstrated very little number sense. Did their answer, the product, make any sense? What would happen to the size of the first fraction as it was multiplied by the second fraction? Was the product bigger or smaller than the fractions? Should it be? We saw students who were simply carrying out some steps without thinking about what multiplying fractions really means.

Continue reading “Multiplying fractions: Why context matters”

New strings for Measurement + Unit Conversion

Our 5th grade team was trying to address several challenges with implementing Common Core 5th grade standards of multiplying and dividing decimals and measurement conversions. Together with our Math in the City coach, Kara Imm, we created these strings.

The purpose of these strings is multi-faceted – we wanted these strings to do a lot, but seem easy and attainable to the students. First, we wanted to use a mathematical model that had the purpose of showing the relationship between multiplication and division, hence a ratio table. We did not want children to see a number, a unit of measure, then wrack their brains trying to decide whether or not to multiply or divide. The model would build habit, confidence, and reasonableness of answer.

Reasonableness of answer was a huge reinforcement of these strings. We felt if students could visualize a cm (say your pinky finger nail) then visualize a meter (arms length) that would push them away from getting stuck on what the conversion rule and help simplify and clarify what operation to preform.

Second, we needed to design the numbers to build off of their prior string work with multiplication and division – so you will see doubling and halving, numbers associated with money. Building upon this foundation of whole number multiplication and division would (hopefully) help us to alleviate some fears of operations with decimals. The ratio table also helped reinforce how to treat the decimal point when working between multiplication and division.

— Nicole Shields, Mariel Simon, Sybil Esenyan, Lori Krellenstein and Allie Minicone (5th grade teaching team, PS 158, Manhattan)

Tips for leading the string: In each case we started the conversation with kids by establishing the unit rate (first line of each ratio table). Then once we had some agreement there, we said, “So if it’s true that 100 centimeters is equivalent to 1 meter, what about 2 meters? Suppose I only was measuring 80 centimeters, now what?” We added values to the ratio table and slowly built the table, using the values here and children’s thinking to guide us. That is, we did not reveal the entire ratio table all at once. It emerged as the conversation did. In some cases we left the last line empty so that kids could make up a new true statement or we could pose a challenging one at the end.

Ultimately, we kept seeing conversion tasks for 5th graders and wanted kids to be able to solve problems like:

  • How many 200 mL paper cups can be filled from a 2 liter jug of lemonade?
  • I make 2.5kg of popcorn and eat 750g of it while watching a movie. How much popcorn is left?

We knew that having the ratio table as a tool to think with would be helpful.








A Dilemma with Models

A 5th grade teaching team I work with recently raised the issue of how to model the problem 1/2 x 3/10 on an array. They saw their students use a variety of models and one teacher got “bothered” by how some of the models felt “imprecise” or “not to scale.” We had a conversation together about this issue.  To prepare myself for the conversation I did a little sketching of possible models that kids might use.  What do you think?

Fraction Models_Multiplication

Halving with early fractions


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.