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:

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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.]

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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…..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

 

Biking on the fractional number line . . .

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

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

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

Multiplying Fractions by a Whole Number

Model: Number Line

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

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

String:

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

(take one or two strategies, helper problem)

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

(take one or two strategies, helper problem)

3/4 of 24

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

1/8 of 24 

(take one or two strategies, helper problem)

3/8 of 24

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

7/8 of 24 

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

5/8 of 24

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

Two notes:

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

 

 

 

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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