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.

Trusting the digits: Developing place value understanding

For several years, across various school communities, a teacher will tell me, “My kids don’t really have a strategy for multiplying decimals other than the ‘stacking’ algorithm.” We talk some about how kids are stacking the numbers to be multiplied, using the whole-number algorithm and then “bumping back” the decimal point to reflect the problem at hand.

“Does the decimal point move?” I ask.

“I think so…” or “Not really, but that’s the idea…” or “Wait! It doesn’t move?” is what I usually hear.

“Could your kids predict the digits in this multiplication problem, without stacking to get the answer?” I wonder.

1.2 x .004

“No way,” they say. And in those moments I developed the kernel of a really promising string, based on the idea of “trusting the digits” and not moving the decimal point. It goes something like this….


Good morning, mathematicians. I know you are working on some decimal operations and today I brought a number string to help us all think about those problems. You know that mathematicians often rely upon a story, or context, as a way to just make sense of what’s going on. Since you are [5th, 6th, 7th] graders, you already know many contexts that we could use. Today, that will be your job — to give us some stories that could help us make sense.

As usual, the number string will start really friendly and then I’ll move us towards problems that will challenge all of us. Ya’ll ready? Got your partner? Okay, let’s go.

Here’s our first problem.

7 x 8 =

I know, I know, we already know the answer. So that’s not my question. My question is what’s a story that would help us make sense of this. And what does the 7 and the 8 mean in your story? What does your answer mean in your personal story? Turn and tell your partner about your context, and then listen to find out about theirs. Go!

After a short turn and talk, I solicit at least three different stories, being sure to record each of them on chart paper.

Okay, so now we have 7 tanks of 8 mini-sharks. Super cool! Thanks for that, Daria. And we have 7 tables of 8 people each, thanks to Rodney. And finally, we have 7 packs of 8 sticks of gum, thanks to Imani. I’m going to record our answer on this place value chart:

Screen Shot 2018-01-15 at 12.33.18 AM

What about now?

7 x 80 =

Let’s take up these stories from Daria, Rodney and Imani to think about: What stayed the same in their story? And what changed?

[Think time, then turn and talk.]

So, what happened in these stories? You can share something you and your partner talked about.

Hector: We talked about how the answer is 560, just ten times more than the last one, but that some of the stories don’t make sense any more.

Can you say more about that? Why is the answer ten times more? What caused that?

Hector: Yeah, so before you had 8 people at a table and now you have 80 people at a table. Ten times more people. But what my partner and were saying is that, that doesn’t make sense — like you wouldn’t have 80 people at a table.

David: But you could flip it.

What do you mean “flip it”?

David: So instead of 7 tables of 80 people, you could have 80 tables of 7 people.

What do you think, mathematicians? And what’s the 560 in their story?

Maria: Number of people all together. At all of the tables.

Okay, sounds like you are saying we might need to modify some of the contexts to make them fit the numbers here, but that it can be done. Other ideas about this?

Franky: Well, I kinda think it’s the same with Imani’s story. It needs a flip.

Who understands what Franky is saying and can build on his idea?

Jackie: So 7 packs of gum with 80 sticks is, like, not really a thing. But you could have 80 packs of gum with 7 sticks in it. Even though, personally, I don’t think they make gum in sevens.

Imani, what are you thinking about this? This was your story….

Imani: Yeah, I think packs of seven would be okay. Kinda small, but okay.

Screen Shot 2018-01-15 at 12.33.33 AMAlright, let’s keep going. Think about this problem — and our stories — and what’s happening to the numbers in these stories. Same questions: What’s changing? What’s staying the same?

8 x 70 =

Seems like lots of you want to check in with your partner? Yeah? Go ahead.

Okay, let’s get a new voice in this conversation — that always helps us. Can someone just get us started with something they noticed? Or something they talked about with their partner? Renny?

Renny: Well, it’s the same but different….The 7 and the 8 basically switched places and the answer stayed the same.

Who can say more about what Renny is saying?

Alina: 7 times 80 is the same as 8 times 70 because they are both like copies of 7 x 8.

Mmmmm….neat! Say more about this “copies” idea….

Alina: They both have 7 x 8 inside of them. And a ten.

Alina, let me try to capture your idea for all of us to make sense of….

7 x (10 x 8) = (7 x 10) x 8

[Depending on the class, the grade level and the goals we have for kids, I sometimes ask kids what this is called. Sometimes the associative property comes up, and when it doesn’t, we just note that.]

Screen Shot 2018-01-15 at 12.33.45 AM

Alright, hold onto your hats for this one. How about 7 x .8?

7 x .8 =

Could any of our stories work here? Why or why not? Do we need new stories for thinking about this one?

[Think time, then turn and talk]

What are we thinking now?

Deidre: None of the stories make sense because you can’t have .8 of a mini-shark or a person or a stick of gum. Right?

So, sounds like the stories didn’t carry over for us in a helpful way?

Let me ask a different question: do you have a story if I do this?

7 x $.80

[lots of “Ohhhs” here] What happened? What’s the “ohhing” about?

Najee: You didn’t say anything about money before. But, yeah, this could work.

Is the dollar sign helping anyone else to make up a story? Let’s hear it!

Kristina: Yep, what about 7 packs of gum and you spent $5.60.

Okay, and where’s the $.80 in your story?

Kristina: My bad. The gums are all eighty cents.

What do we think? Would that work?

Justin: Basically you could make a story where you were buying any 80-cent thing and for some reason you needed 7 of them.

Screen Shot 2018-01-15 at 12.33.57 AM

That’s pretty cool — “any 80-cent thing.” Okay, so how about this one?

8 x .7

I’m hearing murmurs, which usually means a turn and talk is in order. Thirty seconds to check in with your partner. Go!

Marlene, will you share what you and Mariama were talking about?

Marlene: Uh-huh, you could use money again here.

Say more…

Marlene: But now you have 8 candy bars and they each cost 70 cents.

So, does that help you to find the answer to 8 times .7?

Andy: Basically, yes. because you could just add 70 cents eight times and that would give you $5.60.

Hmmm…is that true? Are you all convinced the 70 cents 8 times is $5.60. Lemme record that so that we can see…

70 + 70 + 70 + 70 + 70 + 70 + 70 + 70

          140 + 140 + 140 + 140

                    560 cents

Rodney: Yeah, I’m good. I mean, I’m convinced. Whatever.

Can you say what convinced you, Rodney?

Rodney: I know that 560 cents is the same as 5 dollars and 60 cents.

560 cents = 500 cents + 60 cents

                    $5 + $.60

Mmm-hmmm. Because?

Rodney: 560 is like — 500 cents is 5 dollars and there’s 60 cents left over.

Screen Shot 2018-01-15 at 12.34.15 AM

Okay, so let’s end with this problem:

.8 x .7

[Think time, scanning the room] What happened? Why so many grumpy faces?

Josue: We don’t like this one.

I’m with you. I don’t love it either. Why not?

Josue: There isn’t a good story…..so, like, let’s say you use money. What does 70 cents times 80 cents even mean?

Totally. Well, this is interesting. It sounds like none of us have a great story for this problem — mini-sharks, tables, money, nothing. Be thinking about why that is.

So, let’s pivot away from the story to look at the numbers. Why did I choose these numbers? What do you think is true about the answer, even if you are not totally sure what the answer is? Where is this answer on our place value chart? Let’s turn and talk….

Anyone have an idea about this product? Who can get us started here?

Solomon: Well, we looked the “pink problems” and every single time there was a 7 and an 8 in your problems….and so there was always a 56 in our answer. Sometimes a big 56 and sometimes a smaller 56.

Interesting. Anyone understanding what Solomon is saying — “big 56” and “small 56”? Okay, add on…

Hector: Basically these are versions of 56, where the 56 is just going to the left or to the right depending on how many tens there were. You see? [pointing to the place value chart]

Are you saying that all of these problems has a 56 in it and it’s just a question of where on the place value chart the 56 is?

Hector: Basically, yes.

So, where would this 56 be? How do we use what we know about number to know where to place the 56 on the chart?

Jemma: I personally think of those like fractions, like 7/10 and 8/10 so for me, it’s like 56/100, the regular way, but then you divided by 10 twice.

Okay….and….

Jemma: And that means you move the 56 to the left two times. Divide by ten, divide by ten [gesturing to show the movement of digits to the left].

Let me record this, while someone else chimes in about what Jemma is saying.

Josue: Ooh, so she’s saying that all of these problems are going to be 56, but some are whole numbers — kinda to the left — and others are decimals — kinda to the right.

Josue, here’s a question for all of us, based on what you just said. Is the answer to .7 x 8. here? Or here? And how do we know?

Screen Shot 2018-01-15 at 12.34.33 AMI typically end the string by asking students to think about, write, or share (one of the following):

  • something that got clearer today
  • something they noticed that feels important (and why)
  • an idea someone said that felt important (and why)
  • a big looming question they had

In this string my purpose was to:

  • encourage students to use place value relationships to develop intuition about decimals products — to “trust the 56” in our case
  • support students to “look inside” the numbers to build some confidence about the digits — 1.2 x .004 will result in “some kind of 48,” now we just need to reason about where that 48 will be on the place value chart and why
  • get students to decide/name how one problem was related to another
  • help students to see that the decimal point, in fact, doesn’t move, the digits do — and when they move it means that we are multiplying or dividing by a power of ten

A follow-up string might look like this:

4 x 12

4 x 120

40 x 12

40 x 1.2

.4 x 12

.4 x 1.2

.4 x .12

Thanks to Leslie Hefez (MS 88, Brooklyn, NY), Amy Fitter (Parkway Schools, St. Louis, MO) and Mary Abegg (Hazelwood Schools, St Louis, MO) for feedback and lab-site ideas.

Poster from Leslie’s 6th grade class (and an idea for another string)

image1

 

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 https://teachdomore.wordpress.com/ 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.

Screen Shot 2018-01-15 at 1.41.47 AM

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 [http://www.fosteringmathpractices.com/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.

Cars, coffee, and climbing stairs: Inviting students into the story

Here’s another important contribution from our friend and colleague Marcelle Good — a 6th grade teacher at School of the Future (Brooklyn) and a Math for America Master Teacher.

In this post she illustrates the role of number strings in helping students to reason quantitatively. This idea  — codified as one of eight Standards for Mathematical Practice — means that students of all ages can “make sense of quantities and their relationships
in problem situations…the ability to contextualize.”

It also suggests that students have developed the “habit of creating a coherent representation of the problem at hand….considering the units involved, attending to the meaning of quantities, not just how to compute them.” In other words, giving students the chance to situate numbers and other values in a story and using those stories to make sense of the mathematics.


 

To be honest, I was slow to come to the idea of loving context in number strings — the numbers were so beautiful on their own!  An even bigger issue was that the context never seemed to be taken up by students during independent work or when the problems got more complex.

Recently with my students, though, I came to really appreciate the power of story as a referent for kids. When my entire class tried to convince me that since 100 cars would have 400 tires, 99 cars would have 399, I knew I had a problem.

With just a bare number ratio table, my kids would not have found their way out of this misconception. But, with the context in mind, one of my 7th graders explained to the class, “This is how you think about it: You have 100 cars for some reason. Some jerk comes and steals one of your cars. He doesn’t drive off with one tire — he drives away with four tires, so you have 396 left.”

Marcelle Photo 1

Before another example, some background about me and my students. At my school, School of the Future (located in East New York) I am really struggling to help students access grade-level content. When they enter my school in 6th grade, most of my students are 3 to 4 years below grade level. To address the challenge, our approach has been to take the long view: we are not too concerned with getting them to do 6th grade math in 6th grade. Instead, our goal is to get them to do 8th grade math by 8th grade.

That wasn’t our initial approach. Originally, we tried to teach them grade level content, and scaffold the work by re-teaching or reviewing topics like double-digit multiplication or generating equivalent fractions. What we found, as a school community,  was that this approach was not working, and just not enough. Students were entering 7th grade with a partial understanding of 6th grade, and still lacking a weak foundation.

As a result, I spend a lot of time figuring out exactly what my students know.

Do they have a concept of how the number ten works in addition and why it’s so powerful?

What ideas so they have about multiplication?

Do they have a concept of area?

Do they count only by ones, or do they have some strategies?

Once I know where they are, I work to address their needs in class by meeting them where they’re at. My students who are the least ready for 6th grade material are also assigned to a numeracy class, in addition to their regular 6th grade math class. And in this class, I give myself permission to teach what would be considered K-5 mathematics.

Now, as I prepare to lead number strings where I know the math is more challenging for students, my first question is always, “What story can you tell about what happened?”

Today, the story was that I had to buy coffee for a meeting (this part was true), and that I had bought 4 cups of fancy coffee for $7. The next part of the string was: Then, something happened and I had bought 8 cups. Turn and talk to your partner to tell them a story about what might have happened.

Students said things like, “Then you got back to school and 4 new teachers showed up to the meeting so you had to go back.” We are on the 5th floor of my building, so the students were feeling pretty bad for me.

They groaned when I put a 10 on the chart under the cups of coffee, possibly a sign that they were invested in the context with me. “What story can you tell now?” I asked them. KellyAnn said, “The principal called you and said we had visitors and that we needed two more cups.” Andrew said, “Now we have to figure out what two cups cost, I’m not sure about that.”

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A few takeaways from this lesson:

  • Students who struggled with the math were able to get started on the problem. I’d much rather have students dealing with, “How much for two cups?” then, “What should I do?”
  • This string had a tone of, “What’s going to happen next?” When they found out that I needed 10 cups of coffee now, we started thinking about all the stairs I had to climb (and this became fodder for a new string).
  • The decision to draw pictures of the situation felt authentic and not a tool to use with students who needed “remedial” math education — because we were all imagining it together. There was no point where I felt like I needed to offer a picture, we were just right there in it together.
  • The pictures were a key to transfer. Students who struggled to reason on their own during independent and partner work could be prompted to draw a picture and suddenly they were able to reason through problems.

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One student’s notebook who struggled a lot with proportional reasoning, but then was able to draw pictures to work through the story.

When we think about context in number strings, this question, “What happened next?” gives students an entry point to get started, and often, this translates so quickly to a picture. My students now know they have been invited into a world where maybe we can have a garage with 100 cars, or I can spend an entire day just on coffee runs, and we can wonder about how many stairs we have climbed. After many such invitations, they’re willing to go there with me to think about the math. The numbers are beautiful on their own, but that’s because they tell us a story.

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Feet, inches and yards: Conversions on a ratio table

This post is from our friend and colleague Kathy Minas, a 4th grade teacher at PS 158, and an avid strings enthusiast.


Recently, our fourth grade team met with Math in the City co-director and staff developer Kara Imm, who has been working with our school for several years. We wanted to explore ways to introduce Common Core 4th grade standards of measurement and conversion by using strings. What follows is the string we designed together and my notes about how to lead it with kids.

Why this string? Why a ratio table?

The purpose of this string is to introduce students to conversions, using the most common units of measurement, inches and feet. We wanted to ensure that students had a familiar model that visually captured the (twelve-to-one) relationship between these units, which led us to using the ratio table. Our students have worked with the ratio table before in both our multiplication and division units of study. The model allowed them to represent and maintain the relationship between known units in order to multiply, divide, break apart, or even add groups with ease.

Performing measurement conversions on a ratio table also supported students to monitor the reasonableness of their answers. In addition, it encouraged them to keep the relationship between units in mind.  Instead of memorizing whether they needed to multiply or divide feet to get inches, they simply trusted the existing relationships on the table that we build together.  In fact, student were much more flexible about how to convert and did not rely on a memorized rule or catchy mnemonic to solve these problems.

Tips for leading the string:

We build ratio table together with the class, instead of revealing the entire completed ratio table all at once. We add values to the ratio table — one at a time, increasingly more complex — and ask students to determine the corresponding number of inches or feet. Some values are added to the ratio table as students explain their process.

The string:

Begin by naming a true statement to ground the conversation:

Mathematicians, we know that there are 12 inches in one foot.

Draw a ratio table and label the columns, number of inches and number of feet. If this piece of social knowledge is not known by your students, having a 12-inch ruler to see and touch is also useful at this moment.

Since we know that there are 12 inches in one foot, how many inches are there in 3 feet? How do you know?

Student responses may include:

  • I know that there are 12 inches in one foot, so there are 36 inches in 3 feet because you multiply the number of feet by three, which means you have to multiply the number of inches by three.

Note: In order to move away from additive reasoning on the ratio table  also known as repeated addition or “chunking” I purposely do not record the unit rate of  12 inches in 1 foot on the table until the students bring it up as part of their reasoning. I’m nudging students from additive to multiplicative reasoning on the ratio table. You may wish to ask students to visualize three feet, “What does this look like? What does this make you think of?”

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A fourth grader (and sports enthusiast) in my class introduced the unit of a yard immediately, so I added it as a third column to the ratio table and made a post it note 20151218_160434about his unique contribution. I left the yards column blank for most of the string, but then we returned to it later to reason about this third unit.

Mathematicians, how many feet are there in 60 inches? How do you know?

Student responses might sound like:

  • I know that there are 12 inches in 1 foot, 60 inches is 5 times greater than 12 inches, so 60 inches are equivalent to 5 feet.
  • I know that 36 inches is equivalent to 3 feet, and 60 inches is 24 more inches than 36. 24 inches is equivalent to 2 feet, so I need to add 2 feet to 3 feet and that’s 5 feet.

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So, how many feet are there in 72 inches? How do you know?

Students will likely reason:

  • I know that 72 inches is 12 more inches than 60 inches, which means I need to add one more foot to 5 feet, which is 6 feet.
  • I know that 72 inches is equal to 36 inches times 2, if I double the 36 inches, I have to double the 3 feet.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”  

Okay, how many inches are there in 9 feet? Can you picture this?  What are you thinking?  

I heard students say:

  • I know that 9 feet is three times 3 feet, so there are 108 inches in 9 feet because I have to triple 36 inches.
  • I know that 6 feet is 3 feet away from 9 feet, so I need to add 36 more inches to 72 inches, which is 108.

Follow up question, “How many yards are there in 72 inches or in 6 feet? How do you know?”  

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Once we were done with the inches and feet component of the string, we tackled the yards. We returned to the relationship between feet and yards. I asked students to consider,

If we know that there are 3 feet in 1 yard and we only have 1 foot, what part of a whole yard do we have?

Then I asked,

If we have 2 feet, do we have a whole yard yet? What part of a whole yard do we have?

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Later I said,

Using all you know about the relationship between feet and yards and all you know about fractions, if we have 5 feet, how many yards do we have?

My students tackled this problem with ease:

  • I know that there is 1/3 yard in 1 foot, so there are 5/3 yards in 5 feet.
  • So, there is 1 yard in 3 feet and 2/3 yard in 2 feet, which means that in 5 feet there is 1 2/3 yards. I just put them together to make 5.

 

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Poster from Kathy’s class, end of string

As big ideas or important strategies come up, my colleagues and I have begun annotating the strings poster so that kids can both see and hear these ideas.  I listen carefully for students to make these contributions we write and display them so that the ideas are shared and accessible to all kids, even if they are still emergent.  Sometimes, when students are ready, I nudge kids towards a generalization, which helps us move beyond the specific string and into other related quantities and relationships. Examples of this practice of annotation are below:

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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?”

On the rug with Angela

It’s late in the school year and I’m sitting on the edge of the rug in Angela Fiorito’s 1st grade class at PS 158 in Manhattan. There is no doubt that the kids are excited to begin math, and, in particular, a number string. The class is working on addition, particularly making use of four strategies that were initiated and named after students in the classroom. Continue reading “On the rug with Angela”

Why Conjectures Matter

This post is from our colleague and friend William Deadwyler, a 6th grade math teacher and strings enthusiast who works at MS 22 (South Bronx).

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This summer I met Sylvia Glauster, a 5th Grade teacher at The Ancona School in Chicago.  Sylvia led a summer institute for 5th – 8th grade teachers at Math in the City on geometry, based on a new unit she co-wrote The Architects’ Project. My work with Sylvia inspired me to start gathering and publishing students’ conjectures. Outside of my classroom is a bulletin board where conjectures are published. Some of the conjectures are right, some are wrong. All were generated in class, based on investigations and number strings that we discussed together.  Investigating these conjectures will help students develop the curiosity and persistence that all successful mathematicians share.

Continue reading “Why Conjectures Matter”

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…..”

Beyond Skip Counting

This post comes to us from strings enthusiast and middle school teacher Marcelle Good, who works at School of the Future (Brooklyn, NY) and is a Math for America (MfA) Master Teacher.

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My 6th graders love to skip count. If you were to ask them whether, for example, 6 was a factor of 96, many would be willing to skip count (by sixes) all the way to 96. Even if they don’t find this tedious and inefficient, I do. So I’ve been working this week on generating some faster strategies for finding factors of numbers — also known as divisibility rules. Continue reading “Beyond Skip Counting”

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”

What’s in a name?

This post is by reader and strings enthusiast Laura Bofferding, Assistant Professor, Purdue University

I was first introduced to number strings by Jennifer DiBrienza (one of the teachers highlighted in Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction by Fosnot and Dolk) when we both worked as teaching assistants for an elementary mathematics methods course. Captivated by their complexity and ability to hook students of all ages, I began to use number strings myself with teacher candidates. Now, as an assistant professor at Purdue University, I routinely explore them with my undergraduate mathematics methods students and require them to try a number string in their practicum classrooms. As I have looked for resources and talked with colleagues about this practice over the past few years, I’ve noticed an increased focus on mathematics instructional routines that people refer to as math talks, number talks, number strings, math strings, cluster problems, and problem strings. Some of these things are not like the others…

Continue reading “What’s in a name?”

Moving straight ahead

Those of you who are fans of the middle school curriculum Connected Math Project (CMP) will especially appreciate this string. I was preparing for a visit to MS 22, a middle school in the South Bronx —  my collaborator Erica Berger, a thoughtful and dedicated teacher, asked me to design a string to introduce linear relationships and to prepare students for a potentially messy investigation in CMP.

Some would say number strings are “curriculum neutral” or “curriculum impartial.”  That is, they are not tied to or loyal to any one curriculum — the routine of number strings can be a helpful and grounding experience for all kids. The challenge, for all of us, is locating and/or designing strings that will support the lesson or investigation that follows.

Continue reading “Moving straight ahead”