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.

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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.

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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…, 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.


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)



Number string structure and design

How are number strings designed? Typically, people tend to describe number strings as having the following structure

Entry problem

Helper problems

Challenge problem (or clunker)


This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this:

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Again, the concept of helper problems. But is there just one “formula” for a number string? Continue reading “Number string structure and design”

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.”

Marcelle Photo 2

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.

Marcelle Photo 3

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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Video: Division on the open number line in fourth grade

Wondering what a number string looks like in a classroom? Curious about the details of the routine? The following video features site co-founder Rachel Lambert teaching a group of 4th grade students at the Citizens of the World Charter School in Mar Vista, CA. These students have a wonderful classroom teacher, Hayley Roberts, who does number strings regularly with the students as part of a rigorous, inquiry-based mathematics curriculum. Hayley is off camera in Part 1 leading the students in a mathematics mindfulness exercise.

Students in this class had done lots of number strings with the open number line for addition and subtraction, as well as the array as a model for multiplication. On the previous day, the number string had been a multiplication doubling and halving string on the open number line. Today’s number string was designed to help students understand division on the open number line, focused on using equivalence as a strategy. Before you watch, you might want to anticipate how 4th graders might solve these problems, and how Rachel will represent strategies on the open number line.

2 x 50

4 x 100

100 ÷ 2

100 ÷ 4

200 ÷ 4

400 ÷ 8

800 ÷ 16


Continue reading “Video: Division on the open number line in fourth grade”

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.


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”

Chocolate Arrays

Lately, I’ve been seeing arrays everywhere I go: at the grocery store, at the pharmacy, at the farmers’ market. And, of course, at Costco. The big, bad bulk retailer is bursting with interesting items arranged in perfect columns and rows.

Naturally, I made a beeline for the chocolate.

What follows is a quick-image string for exploring the associative property, the patterns that occur when multiplying, and the relationship between columns and rows. It supports the development of some key strategies for multiplying: doubling and halving to maintain equivalence, doubling a dimension to double to product, and using partial products to solve.

Continue reading “Chocolate Arrays”

The delight of disequilibrium

Disequilibrium is Piaget’s term to describe when what a learner already knows comes into conflict with new information. Learners must work through the confusion to reconstruct new knowledge. How does the process feel to a learner?  How as a teacher can we respond during a number string when students demonstrate disequilibrium?

Continue reading “The delight of disequilibrium”

Searching for Friendly Numbers

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


Continue reading “Searching for Friendly Numbers”

From additive to multiplicative thinking

A few weeks ago our colleague and collaborator Pam Weber Harris led a really interesting numeracy workshop at Math in the City (City College). A former high school teacher and now teacher educator based in Austin, Texas, Pam has expertise in many areas: technology, assessment, K-12 mathematics, and more recently numeracy routines.  Her latest publication, Building Powerful Numeracy for Middle and High School Students, is a direct outgrowth of number strings that extends our work to college level mathematics.  Pam renames number strings as problem strings, but the essence of the routine remains the same.

Continue reading “From additive to multiplicative thinking”

Arrays on the West Side

We recently learned that the third grade team at Manhattan School for Children (Elizabeth Frankel-Rivera, Madelene Geswaldo, Alice Hsu and Marissa Denice) started a really cool Homework Page for their classes.  In addition to reading for 20-30 minutes a night and writing two entries in their ELA homework section, third graders are also expected to think about and solve a set of math problems.

Continue reading “Arrays on the West Side”

Closed to Open Array

After looking over beginning-of-the-year assessments, a new 4th grade teacher was concerned that half of her students were still unsure of the open array model. Some were simply still not convinced of the empty boxes! As her coach, we planned a number string together that would engage the students through questioning, get to know the students even more through open discussions (since it’s still September), and to help each student “hook in” and trust the open array. I modeled this string, hoping to model the questioning but to also model for the teacher who is coming across the open array for the very first time.

Continue reading “Closed to Open Array”

How to extend a string

We are often asked about how we can “stretch” strings beyond their place as a short mental math activity in the classroom.  How can they be used as part of a formative assessment of individual kids? What might come after a string that isn’t necessarily another string? What can kids do at home that builds on the thinking and reasoning that we are developing by doing strings?

Continue reading “How to extend a string”

Making Thinking Visible

I recently worked with a 6th grade teacher, Miss T, as she led a string for the very first time.  Twenty-eight middle school students quietly re-arranged desks and chairs and situated themselves at the front of her room — itself, no small feat — as she prepared to facilitate a messy multiplication string:

17 x 10 =

17 x 2 =

17 x 12 =

17 x 20 =

17 x 19 =

17 x 21 =

Continue reading “Making Thinking Visible”

Photo number strings for multiplication

Here are two photos I snapped as I walked by a 99 cent store in LA. Beautiful arrays, no?  Image

I am thinking about how to use these kinds of images as the anchors for number strings, particularly for intervention work with older students.  Sometimes older kids need work thinking about multiplication, but in an age-appropriate way.  What kind of questions do you think of with this image?  One could most simply begin by asking what kids noticed about the image.  That would bring most of the interesting mathematics forward, I think. Beginning perhaps with how many boxes of hot chocolate do you see (nice numbers)?  And then, considering this is a 99 cent store, how much would it cost to buy all of this chocolate.  It reminds me of some work that Pamela Harris suggests in her book on Powerful Numeracy, in which she asks kids what is 99 plus any number?  A 99 cent store is a great way to think about what is 99 times any number?

Continue reading “Photo number strings for multiplication”