# Ep 138: To Include or Not: An Important Subtraction Pitfall

February 07, 2023 Pam Harris Episode 138
Math is Figure-Out-Able with Pam Harris
Ep 138: To Include or Not: An Important Subtraction Pitfall

Do you ever fall into the "off by one" trap? Do your students ever get confused about counting objects vs counting distance? In this episode Pam and Kim use a Problem String to help build intuition about everyday subtraction problems.
Talking Points:

• Experience a Problem String to build relationships
• Fencepost Problems are all over the place
• Students can't just pluck numbers out of Fencepost problems
• Cathy Fosnot's replacement unit, "Measuring For the Art Show"
• Other places these kind of problems show up
• Kim explains her strategy

See episode 111 for more about "Measuring For the Art Show"!

Check out our social media
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education

Pam Harris:

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague:

And I'm Kim.

Pam Harris:

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians, not only our algorithms and step by step procedures, not particularly helpful in teaching mathematics. But rotely repeating those steps actually keep students from being the mathematicians they can be.

Kim Montague:

So we get some great ideas from all over the place about what to chat about on the podcast. Right? And so this week, we thought we would talk about a tweet that we actually saw quite a while back, but it's popped up in conversation between us several times, as we talk about subtraction. So Pam, we've actually joked about this a few times, because sometimes as we talk about this particular type of problem, one of us will go, it's this right?

Pam Harris:

Kim Montague:

Okay, so I would expect that some students would think about that like a subtraction problem and say 23 minus 17. And then, you know, they could solve it a couple of different ways. So they might start with 23 on the number line. And you know, I like Over, so they might subtract 20 to get to three, and then realize they removed too much. So then they add three back on to get six, six pages. Somebody else might say. "What is the distance between 17 and 23?" And so I have to read six pages, because that's the kind of the gap between 17 and 23.

Pam Harris:

So you think 17 up to 20? Twenty up to 23? That's the two threes and that's six. That's kind of, and the context kind of calls for that kind of looking between 17 and 23. Cool. So as a teacher, we would probably represent one or at least the distance one, maybe the removal one, but we would expect students to kind of be thinking about one of those two, and we'd say, "Great." Okay, next question. There are some parking spots in the senior parking lot left to paint. So I don't know if you guys have that at your school. But sometimes schools will like allow senior students to 'rent' a parking spot for the year and they can go paint it and make it their own or whatever. So there are some parking spots left to paint in the parking lot. How many more kids can choose a spot if spots 17 through 23 are open?

Kim Montague:

Hmm. What should I expect students to do?

Pam Harris:

Umhum.

Kim Montague:

I would expect that many students will count by ones and say, "17, 18, 19, 20, 21, 22, 23."

Pam Harris:

And when you just counted that, how many did you get?

Kim Montague:

Seven spots.

Pam Harris:

But I thought 23 minus 17 was six. Didn't we just decide that the distance between 17 and 23 was six?

Kim Montague:

Yeah.

Pam Harris:

But you just got seven. There's seven spots.

Kim Montague:

Yep.

Pam Harris:

That's interesting. And that's literally what I would say in class. I would be like, "That's interesting. Wait, what?" And I might ask students, "How are you making sense of that?" But I don't want to kill it too much in this first, these first two problems. But I do want to encourage students, you know, "How are you making sense of the fact that both of these problems have a 17 to 23 in them? And for one of them we got the answer six, and for one of them, we get the answer seven." Next question. What's the distance between the eight centimeter mark and the 13 centimeter mark? Now, ideally, I would maybe show a picture where I would have like, we call it the broken ruler problem, where you kind of see like this jaggedy edge where you don't see the rest of the ruler, and you kind of see maybe the seven centimeter mark and eight centimeter mark, and then 9, 10, 11, 12, 13 centimeter mark, maybe the 14 centimeter mark. So if I'm asking between eight and 13, I might have a centimeter on either side. And I might then draw a line on top of the ruler that kind of shows I'm looking between eight and 13, the eight centimeter mark and 13 centimeter mark. So the question would be, what is the distance between the eight centimeter mark and the 13 centimeter mark? This is a typical question that showed up on the NAEP exam. It's one of the things that we talk about a NAEP exam, we call the broken ruler question. So Kim, what would you expect students to do with that question?

Kim Montague:

It was also a very common question in third grade state exams for Texas, years back. So between, I would think many students would say, "Okay, between eight and 13, I might figure out between eight and 10 will be two centimeters. And between 10 and three would be three more centimeters...

Pam Harris:

Ten and 13. You said 10 and three. Between 10 and 13.

Kim Montague:

So the total distance between eight and 13, would be five centimeters.

Pam Harris:

Those five centimeters. Yeah. And hopefully, you know, like, you can picture that. We would probably draw an open number line on the board and maybe do those jumps, the 8 to 10 and the 10 to 13. Cool. All right. Next question. How many mile markers, so we're driving down the road, and there's these mile markers. How many mile markers, those are signs. So I'm aware that we have international listeners and so like signs that every mile on highways often will have markers of how many miles you are from the beginning of that whatever the beginning is, often it's the state line. So how many mile markers do you need to replace, if mile markers eight through 13 got blown over in the storm?

Kim Montague:

Yeah. So again, I would expect that many students would think about each one of those individually.

Pam Harris:

Okay.

Kim Montague:

And so they might say, "Eight, 9,10, 11, 12, 13. So that's six mile markers."

Pam Harris:

Because you have to replace mile marker eight. You have to replace my marker 9, 10. So when you were like saying, like you said, counting by ones are actually counting each of those mile markers. So you're saying that even though we had eight and 13 in both of those questions that the distance between the eight centimeter mark and the 13 centimeter mark was five, but when it was the eight mile marker to the 13 mile marker, then we had to replace six?

Kim Montague:

Yeah.

Pam Harris:

Like why do we have five one time and six another time? That's confusing. I have one more question. How many fence posts do you need to paint, if you're painting fence post number 34 through fence post number fourty.

Kim Montague:

Okay, so this is going to be one like the one you just asked me where I have to consider that I'm painting the 34th fencepost.

Pam Harris:

Okay.

Kim Montague:

So, I'm gonna paint each one of those 35, 36, 37, 38, 39, 40. What did I just say, 34, 35, 36, 37, 38, 39, 40. I was like wait, why did I get 6 not 7. That's weird. So there's gonna be seven fence posts.

Pam Harris:

So when you recounted you got seven?

Kim Montague:

Yeah.

Pam Harris:

Even though 40 minus 34 is clearly six.

Kim Montague:

Yeah.

Pam Harris:

Thirty four plus six is clearly 40. But you're saying the answer is seven. So, listeners, you might want to actually like draw the fence posts, like countless fence posts, are there really seven fence posts. Okay, okay. Hey, what's the distance, how much wire do I need to put how many feet of wire if they're a foot apart, that's dumb a meter apart. I don't know how much fence posts or how much wire you need, but the distance between them from 34 to 40 is six. The distance between the six, whatever that unit is. So if the fence posts are, say this is a really skinny fence, I don't know, the fence posts are very close together. So 34 to 40 is 6. But if I'm painting fence posts, 34 to 47, fence posts, and we call this fence post problem. At least I do, I don't know why. And so I wanted to end this string with fence posts, so that I could start calling it the fence post problem. This is what Kim and I will refer to, we'll get to some sort of problem in lots of different contexts, lots of different situations, higher math, lower math, you know, like all over the place where all of a sudden, I'll go, "Ah, this is a fencepost problem." Where all of a sudden, I realize it's not just a distance between problem. But it's a fence post problem. I can't just look at how far apart the posts are, I have to actually count the posts. So a big takeaway, a big question I would ask students is, "Are you counting objects? Or are you finding the distance between objects?" And this is a... go ahead.

Kim Montague:

Well, I was gonna say I'm struck by how this is exactly the scenario or the situation, why it's so important that we don't have kids just pluck numbers out of a problem. Because clearly the context of the situation is changing your solution.

Pam Harris:

Kim Montague:

No, I was thinking if there is anything I wanted to say. But I don't think anything I want to add.

Pam Harris:

No problem. So listeners, Kim and I were talking and she's like, "Sometimes I don't feel like interrupting you, Pam." And I was like, "Oh, then I'll pause more often." She's like, "We'll just pause and let me hang there." Okay, I don't know. [laughs]

Kim Montague:

[laughs] I'll jump in.

Pam Harris:

So I was actually talking about this with one of my sons the other day, and he said, "Oh, yeah, like, this is a thing in computer science." And I was like, "Tell me more." And he goes, "Oh, it's often that will have strings in computer science." A string is like a list of characters. So it could be numbers. It could be letters, it could be words, but you know, it could be names. So you have this list of characters, and then there are times you need to do things with those. And you have to decide, are you going to be inclusive? In other words, are you counting fence posts? Or exclusive where, no, you're leaving out that first fence post? And you're just looking at the span or the distance between them. And you have to make sense of that. And it was interesting for me that that's kind of where it came up for him was in computer science. All of a sudden, he had to make sense. Do I, what do I mean? Do I mean the span between the fence posts or the tick marks? Or do I mean that no, no, no, I actually have to count that first fence post as well. And so he said, like, sometimes you'll have to order lists, or edit lists and string and you have to know the number of objects, not the distance between them. And so that's a huge, that's a specific instance. But then Kim, you were like, oh, but Pam, it shows up everywhere.

Kim Montague:

Yeah, I think it shows up in our daily life more than we probably realize. And now that we're mentioning what we call fence post problems. I think listeners will probably be able to go, "Oh, wait, that that is a fence post problem." And I'm wondering if people might find themselves kind of reverting back and counting in those situations?

Pam Harris:

I mean, I did.

Kim Montague:

Yeah.

Pam Harris:

I did so often.

Kim Montague:

Yeah, because it's just so it's just like a little twist on a subtraction problem that maybe shakes you up a little bit.

Pam Harris:

Oh, and I got really clear that if I, once I realized it was a fencepost problem, if I didn't count, I was getting it wrong. And I was trying to get it wrong. And so I was like, I'm just gonna count. Then I know, I'm right. And what I didn't do was take the time to compare. What I didn't do was take the time to compare. So let's look back at the problem string that we just did a minute ago. So I said, "We're on page 17. There's 23 pages in the book." You said, how many pages were there left to read?

Kim Montague:

Six.

Pam Harris:

Six. So that's like 23 minus 17. Yeah? Then I said, "Okay, but they're senior parking spots, number of parking spots 17 through 23." And you said the answer was?

Kim Montague:

Seven.

Pam Harris:

Seven. So if you're just subtracting and looking at span, the answer was six. If you're counting objects, counting spots, the answer was seven. Is it always one off?

Kim Montague:

You wanting me to say something?

Pam Harris:

I kind of am, yeah.

Kim Montague:

Okay. Yeah. So yeah, it's one because, and I don't know when I generalize this, but I think about it like, it's the distance plus one. And so I'm going to go to your fence post, I'm going to jump ahead to your fence post problem and give kind of a picture in that context. So if I'm standing at the first fence post, or the 17th, or whatever number you gave us, then you can think about the span of it, like the space and the fence posts, this space and the fence posts, the space and the fence posts and keep doing that till you get to the last fence posts. So there's that kind of span. But then you've got to remember to come back and pick up that first fence posts. So you can think about it like the span or the distance between them. But oh, yeah, don't forget that first one.

Pam Harris:

And the reason you can think about it, this is span or distance because as you said, as you're walking down the fence plus and I'm gonna actually use the numbers we used in the string so 34 to 40. So if I start at the 34th fence post, but I'm not counting it, because I'm thinking about distance, then I'm going to count that span from 34 to 35. That's also equivalent to counting the 35th fence post. Then I count the span from 35 to 36. That's like I've grabbed the 36th fence post, 36 to 37. So as I count each of the spans, I'm also kind of grabbing the fence posts that goes with it. But I've missed the first fence post if I do it that way. So you're saying I can just count the number of spans because that's equal to the number of fence posts. But then I'm going to add one more, because I would have missed the first fence post.

Kim Montague:

Yeah.

Pam Harris:

So y'all, we can help students realize that in fence post problems, we got to go back and grab that first fence post. So if I would have asked you, Kim, a question like, how many mile markers do you need to replace between eight and 13? What would you really have done?

Kim Montague:

I would have thought about the distance between eight and 13, which is five and then added one more. So it's six.

Pam Harris:

Bam if it was six. That's your strategy. Listeners, I wonder what your strategy is? Hey, y'all, thanks for tuning in and teaching more and more Real Math. To find out more about the math is Figure-Out-Able movement visit mathisFigureOutAble.com. Let's keep spreading the word that Math is Figure-Out-Able.