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)

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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.

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

For my first period class I planned this string

13.2 ÷ 1.1

3.6 ÷ 1.8

7.2 ÷ 4.5

245 ÷ 3.5

32 ÷ .25

3600 ÷ 1.4

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

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

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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)

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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.

New strings for Measurement + Unit Conversion

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

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

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

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

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

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

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

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

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

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