## Differentiating Number Strings: The BandAid Problems

How do you plan a number string for a group of learners who are working on very different skills in math? This last spring my pre-service teachers at Chapman University were tutors in a Math Club. Our club was hosted by a school for students with learning differences, and our students ranged from 5th to 8th grade. In terms of their knowledge of operations, our students were at very different places. Some of the students were working on understanding addition and subtraction with smaller numbers, while others were able to quickly compute using all operations with both whole and rational numbers. Each week we started with a whole group math activity, then students would work with their Chapman pre-service tutors.

We work on number strings in class, as a routine that can accomplish multiple goals. Two of my students, Stephanie Weinfeld and Tayler Martin, planned a number string on constant difference in subtraction. They took up the challenge of differentiating this number string in several, interconnected ways. They provided context, models, and made children’s thinking explicit. Below is their narrative . . .

Differentiating a number string for students who are struggling in math can be tricky. Due to the range of levels in the math club, differentiation was crucial. Some students were beginning to understand adding and subtracting, while others in the club were multiplying and dividing multi-digit numbers. This got us thinking, “How can a number string be beneficial and engaging for ALL of the students present?”

The first step in making this number string a little different was making sure that students were ready to see the numbers using different models. We started with an estimation game with band-aids in little plastic jars, showing the number 35 in different ways.  After Band-Aids per jar were revealed we posed the question, “when you think of 35, how do you picture it?” Students were given time to discuss this question with each other. In a large group, a few students shared their strategies while we illustrated them on the board for future reference.

Most students shared that they, ‘just pictured the number 3 then the number 5’. With some prompting, some expressed that they imagined Snap Cubes (manipulatives) in 3 ten blocks and 5 ones. In order to provide students with as many visualizing strategies as possible before jumping into the number string, we presented 35 on a number line. The visualizing chart proved to be highly beneficial later on in the lesson when differentiating a number string.

Our second step to differentiate the number string was incorporating contextual word problems. We shared a Shel Silverstein poem about band-aids with the students. Students used connecting cubes to model this, which provided exactly what they would need later- a model of the number 35. Instead of just listing the problems, we made slides with a context, ripping bandaids off a person. The word problems provided a tool where students were really able to visualize the removal of Band-Aids while having fun.

This constant difference number string began with a special number; you guessed it, 35!

Number string:

35-20
30-15
34-19
44-29

Introducing manipulatives in a non-threatening manner made students feel comfortable no matter which strategy they used. We kept the routine true to a typical number string by not suggesting that students take out paper. This was so students had to think outside the box and use the tools around them to solve this number string.

Whole group number string activities support many types of learners because multiple peer strategies are shared and illustrated continuously throughout lessons. Each equation was presented in number string form, individually and sequentially. In addition, this number string was given with corresponding word problems. To provide context, all word problems were written about Chapman tutors and Band-Aids. After reading each problem aloud, students were given ‘think time’ to find an answer.

The goal of this string was to introduce constant difference on a number line. Much like the lesson thus far, a pattern emerged. Initially, all of the students that were sharing their strategy used the standard algorithm to solve. In an attempt to steer students away from the standard algorithm, we then asked them to turn and talk about how they could represent each equation on a number line. After discussion and providing visuals, some students began to imagine these problems on a number line.

Some of the students are most successful when using manipulatives, like Snap Cubes, to solve. However, because there is a wide age range in the club, many students do not want to be the first to use blocks to solve. In order to introduce Snap Cubes in an open and relaxed way, we directed students’ attention to the visualizing chart. We said, “I picture this problem in cubes!” This lead to a discussion about tens and ones blocks, and different ways the class could visualize breaking apart the blocks to find the solution. Students began to use different methods to solve each equation. All around the classroom, we could see some kids using blocks, others a mental representation of constant difference.

Instead of the number string focus remaining constant difference, it became visualizing and strategy practice. As a group, after the last problem of the number string was completed, students collectively named each strategy. The names the class awarded the strategies were line hops, stacking subtraction, block subtraction, and adding to subtract.

We took a fairly simple number string on subtraction and added in multiple entry points for kids. Some kids needed a story to hook them into the problems. Others enjoyed the humor about band-aids and the Shel Silverstein poem. For some kids, they needed the tangible, concrete support of blocks to model their thinking. But ALL kids participated in mathematical discussion, and ALL kids named and identified multiple strategies for subtraction.

## The Mathematics of Core Sampling: Division of Fractions

Keely Zaientz and Corey Levin teach 6th and 8th grade math in an integrated co-teaching classroom at Yorkville East Middle School. They have a progressive, constructivist classroom, centered around students developing and identifying their own strategies to approach problem solving. They are huge fans of number strings and spend a lot of their time trying to get better at teaching.

Number strings have become an essential part of our classroom culture. We frequently use them to launch a new unit of study or to reinforce a topic that students have started exploring, but have misconceptions that need clarification.

Our students frequently struggle with the meaning of fraction division. This is a topic that makes a great deal of sense intuitively; however, when students need to identify strategies to support their intuition it seems to violate some unwritten rule about fractions that just makes it so confusing. This fraction string was developed to utilize models to support an understanding of what fraction division means before introducing students to the notation of fraction division.

At the start of each string, we gather the students close to the board and have them bring their notebooks and pencils. This allows them to stop and jot their thinking as situations are discussed.

For this string, I (Keely) share that my brother is a geologist and we are going to explore a tool that geologists use called a corer. We show them some pictures of the coring process and ask students to identify their noticings.

We usually look for noticings like, “Part of the corer is in the water and part is out of the water” or “It is taking a measurement of some sort.” These observations get us closer to being able to mathematize the work of the geologist.

We often provide some background information as well, like the idea that it is so cool that geologists can determine how the environment has changed over many hundreds of years simply by taking core samples. A core sample is collected with a corer that you put into the sediment and it pulls up buried layers of wet mud in the order of it settling. The sampling that is going on is happening in a relatively shallow part of the river, near the banks. Geologists use a corer to bring up buried mud in an effort to study it. Once we feel as if our students have an initial understanding of this context, we are ready to introduce some values for them to model and eventually make sense of.

The first situation goes like this …

At the core sampling site, I noticed that there was a geologist with a corer that was half in the water and half out of the water. The part that I could see in the air was 2 feet long. How long was the corer?

Students have some time (about 1 minute) to discuss the situation with their peers and are asked to model their thinking.

We share how solutions were identified and we show one model of the situation on the board. Frequently, one student draws a corer partially in the water, and out of the water with 2 feet labeled on both parts. If a student doesn’t volunteer this information, we often try to solicit it from students or co-construct a sketch with our students, so that we all have one clear model. This model stays on the board and becomes a scaffold to all learners.

Here are some pictures from student notebooks and the noticings that we have jotted on the board:

In the second situation …

I got to the core sampling site, I noticed that there was a corer that was ⅓ in the air, 1/3 in the water, and the rest was in the mud. I was told that the part in the mud was 2 1/3 ft long. How long was the corer?

At this point, we usually allow students to take one minute to jot their own thinking followed by 2 minutes with a partner to develop a model that will tell the story. If students finish early, we are ready to challenge them to model their thinking with an equation.

Again, we have a group discussion about the models that were developed and any ideas they have generated about developing an equation. It is typical that students identify 2 feet by 3 parts of the corer will give you 6 feet, but will be unable to use 1/3 in their model. We allow our students to develop hypotheses related to how they deal with the extra foot and how they “split up” the numbers that they are interacting with each other.

The student notebooks often look like this:

In the final situation …

The best core sample was about to happen. I could tell by the corer they were using. The corer had 3 ft out of the water. I was told that 1/2 of the corer was in the water and ⅓ of the corer was in the mud. How long was the pole?

Again, we use the routine of having students think about it for one to two minutes and then talk with a partner. With some classes this takes 5 minutes and with others it can take 10 minutes or more. Again, if some of your class finishes early, challenge them by developing an equation to represent the problem and forcing them to prove why 3 divided by 1/6 is 18 feet.

The discussion for this situation usually focuses on dividing the corer into sixths and why 3 divided by 1/6 is the same as 3 times 6.

It is not unusual for students to confused the idea of 1/2 or 1/3 of the pole with half or a third of a foot.  Typically, though, a model of the situation (that resembles a vertical open-double-number line) can help rectify this confusion. Seeing the proportional relationship between the fractional amounts and actual amounts sets our students up well for the upcoming work of proportional reasoning in 6th grade. In this string without telling our students that we are dividing, we offer a situation in which division is about rate and ratio (partitive or fair sharing division). We are nudging our students to associate two quantities (with two different units) in the form of a ratio: number of feet with portion of the entire corer. They will come to trust that this is division, even though many will not recognize it as fraction for awhile.

When we have a group that is really excited about the problem and ready for a challenge, we offer them this extension, which we now offer to you, too:

In this situation, the corer had twice as much length in the water as in the air. There was three times as much length in the mud as in the air. The corer was a total of 14 feet long. How long was the section in the air?

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:

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

Alright, 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.

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.]

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.

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…

## 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.

## \$5 + \$.60

Mmm-hmmm. Because?

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

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?

I 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)

## 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.

## 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.

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

## (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.

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.

After 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)

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

Getting rid of the decimal (Ming)

Looking inside the numbers for common factors (12), then scaling up by 10 (Dante)

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

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

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.”

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.”

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.

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.