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Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Alright. So picture this. You're looking at a stack of quizzes, red pen in hand, and then bam, one score is just way out there. An outlier.
Speaker 2:Yeah. Those outliers. They're everywhere. Right?
Speaker 1:Not just in our classrooms. Mhmm. But for us teachers, figuring out those outliers in student data, it's kind of a big deal.
Speaker 2:It is.
Speaker 1:It's about more than just the math. It's about understanding what's going on with our students.
Speaker 2:Mhmm.
Speaker 1:Right. So today, we're doing a deep dive on lesson 14, outliers from illustrative mathematics algebra 1 curriculum.
Speaker 2:Oh, outliers. Nice.
Speaker 1:By the end of this deep dive, you'll be totally prepped to guide your students through this whole world of outliers.
Speaker 2:What I find really interesting about this lesson is that it takes something like outliers, which sounds, you know, maybe a little niche.
Speaker 1:Right.
Speaker 2:But it shows how it applies beyond just algebra class.
Speaker 1:Totally. Like, data literacy for everyone.
Speaker 2:Exactly. Yeah.
Speaker 1:So let's start with the basics. What exactly is an outlier?
Speaker 2:Well, and this lesson doesn't just say, oh, it's that number that's, you know, way out there. It uses the interquartile range or IQR.
Speaker 1:Okay. I'll be honest. IQR, that always sounds a bit intimidating.
Speaker 2:It does.
Speaker 1:Can we break that down a little bit?
Speaker 2:Yes. Absolutely.
Speaker 1:Make it a little less maybe.
Speaker 2:So imagine you've lined up all your students' test scores from lowest to highest. The IQR is basically the range where that middle 50% of your class falls.
Speaker 1:Okay. So it gives you a sense of that typical spread.
Speaker 2:Yeah. Exactly. And then they actually give you a handy dandy formula in the lesson. So anything that falls more than 1.5 times the IQR above the upper quartile or below the lower quartile, that's when it's flagged as an outlier.
Speaker 1:So it's like drawing a line in the sand around what's usual, and then anything outside of that is, like, hold on a second. Outlier alert.
Speaker 2:Exactly. And here's where I think it gets even more interesting, especially for teachers. They make a really good point that outliers aren't always bad data.
Speaker 1:Right.
Speaker 2:It's like that student who just aces a test they were really nervous about. Their score might be an outlier, but it's a good outlier.
Speaker 1:Right. It's about understanding the story behind the data. Right. Right?
Speaker 2:Yes. Exactly.
Speaker 1:Especially when we're talking about our students. So how does this lesson plan help teachers actually teach this? What do they do?
Speaker 2:Well, I think what's really valuable about it is it gets teachers thinking about it from the student's perspective.
Speaker 1:Okay. I see there's an activity here called health care spending. Tell me about that.
Speaker 2:So this is a really good example of how they do that. They use this real world data on health care spending, which, you know, very relatable Right. To introduce this whole idea of outliers.
Speaker 1:So instead of just throwing the IQR formula at them, they're seeing it in action with real numbers.
Speaker 2:Exactly. And this is where they get to use that formula, that q three plus 1.5 times the IQR.
Speaker 1:Okay.
Speaker 2:But it's not just about plugging in the numbers. Right? It's about understanding what it actually means.
Speaker 1:Right. Right. And then that leads perfectly into this next activity here, investigating outliers.
Speaker 2:Yes.
Speaker 1:So tell me about that one.
Speaker 2:So in this one, this is where students really get to calculate things like the mean and standard deviation
Speaker 1:Okay.
Speaker 2:With and without the outlier present.
Speaker 1:Oh, interesting. So they really see how big of a difference that one outlier makes. In this case, US health care spending.
Speaker 2:Exactly. Exactly. And I think it drives home that point that it's not just interesting. It actually has weight.
Speaker 1:Yeah. It matters. Yeah. So we've identified the outliers. We've seen the impact.
Speaker 1:What do we do with them?
Speaker 2:That's where I think this lesson takes a really insightful turn.
Speaker 1:Okay.
Speaker 2:Because, you know, the assumption is that you just get rid of them. Right?
Speaker 1:They're bad. They're messing up our data. Get them out of here.
Speaker 2:Exactly. Exactly. And they're like, hold on a second. And they actually use these really memorable examples, and I think they would really resonate with teachers as well.
Speaker 1:Okay. Like, what? Give me an example.
Speaker 2:Well, so one of them is looking at dice rolls.
Speaker 1:Okay.
Speaker 2:And imagine a student rolls a 20.
Speaker 1:Okay. That's clearly wrong.
Speaker 2:That's wrong. Right? Like, you cannot roll a 20 on a standard die.
Speaker 1:Right. Outlier alert, big time.
Speaker 2:Exactly. And it's just highlighting how sometimes they're just incorrect data. But then they go into an example that's not as obvious. They talk about the number of siblings.
Speaker 1:Okay.
Speaker 2:Because if you have a student who has, like, 12 siblings, yeah, that's gonna be an outlier in your classroom probably. Yeah. But is it impossible? No.
Speaker 1:Right. It makes you think, is it really wrong, or is it just unique?
Speaker 2:Exactly. Exactly. And it makes you consider the context Right. Because it might be an outlier in one scenario, but in another, it's totally valid.
Speaker 1:Totally.
Speaker 2:And then they have one more example that I think is really, really good. They talk about if you're doing, like, a science experiment Okay. And you have a group that has this biodiesel yield that is just way, way off the charts
Speaker 1:Okay.
Speaker 2:Is it a mistake in their procedure, or did they stumble upon something really cool?
Speaker 1:Right. It's not just about finding them. It's about figuring out what they mean.
Speaker 2:Exactly.
Speaker 1:And I love that this lesson is encouraging that.
Speaker 2:Yeah. And that's where I think the real learning happens.
Speaker 1:Absolutely. Now I have a feeling that even with all of this, students are still gonna have some questions.
Speaker 2:Oh, I'm sure.
Speaker 1:There are always some misconceptions when it comes to this stuff. What are some of the things that you've seen? Well,
Speaker 2:even with the best lesson plans, you're gonna run into some snags.
Speaker 1:Of course.
Speaker 2:And one of the things that they talk about in this lesson is that sometimes students just completely miscalculate that outlier formula.
Speaker 1:Oh, yeah.
Speaker 2:You know, remember order of operations.
Speaker 1:Yes. Please excuse my dear aunt Sally.
Speaker 2:Exactly. That can trip them up here too.
Speaker 1:It can. It totally can.
Speaker 2:And so just reminding them about that is really important.
Speaker 1:So even though the formula itself seems straightforward, it can still be tricky. Yeah. It's all about the details. What other misconceptions did they point out?
Speaker 2:Well, the other big one, and this goes back to what I think is so insightful about this lesson, is students just automatically assume that all outliers are bad data.
Speaker 1:Right. Outlier equals air. Get rid of it. Toss it out.
Speaker 2:Exactly. It's messing everything up. It's gotta go.
Speaker 1:Yeah. But we know it's not that simple.
Speaker 2:It's not.
Speaker 1:So how can teachers help students move beyond that misconception?
Speaker 2:Well, you have to go back to emphasizing the context and that, like, detective mode thinking.
Speaker 1:I like that detective mode. I love it.
Speaker 2:Yeah. Really questioning. Why is this an outlier? Could it be an error in how we collected the data? What is this outlier actually telling us about this whole dataset?
Speaker 1:Right. Because it's all about the story. Right?
Speaker 2:It is.
Speaker 1:So are there specific tips that this lesson gives about how to teach this? Because teachers are busy. Right? So how do you fit it all in?
Speaker 2:Yeah. And they do a really good job in this lesson of really spelling out the goals. Okay. And one of the primary goals is that students are able to articulate the impact of the outlier. Okay.
Speaker 2:So remember how we talked about calculating the mean and the standard deviation with and without? It's not just about doing that as a procedural thing, but it's really getting students to understand how much one single number can change things.
Speaker 1:Right.
Speaker 2:You know? It's like I don't know. It's like when you're trying to take a group picture, and there's that one person who's in the back doing something totally crazy
Speaker 1:Yes.
Speaker 2:Stealing the show.
Speaker 1:Yes. Outliers can be like that. Yeah. They can. Totally.
Speaker 2:And they change how we view the whole picture, which in this case is the whole dataset.
Speaker 1:Right. Exactly.
Speaker 2:And I think this lesson really allows students to make that judgment call.
Speaker 1:Okay.
Speaker 2:Like, do we keep this outlier, or does it need to be benched for a little bit?
Speaker 1:It's like giving students a crash course in ethical data analysis.
Speaker 2:Right.
Speaker 1:And this is algebra 1. Right?
Speaker 2:I know. It's impressive.
Speaker 1:That's impressive. Yeah. So from your perspective, you've seen a lot of research. You've seen a lot of data. What did you think was insightful about this particular lesson from illustrative mathematics?
Speaker 2:I think what they do really well is they don't just focus on the what, but also the why.
Speaker 1:Okay.
Speaker 2:So students aren't just memorizing formulas, but they're really understanding what those formulas represent and how to use them in a way that sense.
Speaker 1:Right. And how to think critically about them.
Speaker 2:Exactly. And, you know, they even have a whole section dedicated to potential student misconceptions, which I always love.
Speaker 1:Right. So they're anticipating those questions. Right? So what were some of the things that they flagged that we haven't talked about yet?
Speaker 2:Well, we talked about that whole all outliers are bad data trap. Right. But they also talk about how even on that technical level, students can make mistakes when they are calculating the IQR and that outlier formula.
Speaker 1:Yes.
Speaker 2:Like, it seems very straightforward, but they might mess up the order of operations or just make a careless error.
Speaker 1:Yes.
Speaker 2:You know, it's like I don't know. It's like when you follow a recipe to a tee, but for some reason, your cake is still flat. Flat.
Speaker 1:Right. Those little things.
Speaker 2:It's the little things that can sometimes make or break it.
Speaker 1:It's so true.
Speaker 2:Yeah.
Speaker 1:And it reminds me of a time
Speaker 2:I was baking with my kids, and we were following this recipe. Right? But for some reason, our cookies just wouldn't spread. They stayed in these little mounds.
Speaker 1:Oh, no. What happened? Did you figure it out?
Speaker 2:Well, after much head scratching, we realized we'd use baking soda instead of baking powder.
Speaker 1:Oh, easy mistake to
Speaker 2:make. Right. But it just goes to show, even when you're working with what seems like a simple formula, those little details can really trip you up.
Speaker 1:They can. Okay. So with this outlier formula, we've gotta make sure our students are paying attention to those details. What else did the illustrative mathematics folks flag as potential pitfalls?
Speaker 2:Let's see. Oh, they pointed out that students might struggle with applying the outlier formula to different types of data. You know, this lesson focuses on health care spending, which is numerical data.
Speaker 1:Right.
Speaker 2:But outliers can pop up in categorical data too.
Speaker 1:Oh, that's a good point.
Speaker 2:Like, imagine you're looking at a survey of favorite ice cream flavors.
Speaker 1:Okay. Yeah.
Speaker 2:And most people pick chocolate, vanilla, strawberry, the classic.
Speaker 1:The usual suspects.
Speaker 2:Exactly. But then you have a few responses for, like, pistachio or rum raisin.
Speaker 1:Okay. Those would definitely be outliers.
Speaker 2:They would, but we can't calculate the IQR for those.
Speaker 1:Right. Because they're not numbers.
Speaker 2:Exactly. So the illustrative mathematics folks suggest giving students opportunities to grapple with outliers in different contexts with different types of data.
Speaker 1:That makes a lot of sense. So they're not just stuck in that numerical data box. Okay. So we've talked about some potential misconceptions, but let's zoom out a bit. What are some key takeaways from this lesson plan for our listeners?
Speaker 2:I think one of the biggest is this idea of not just teaching students what outliers are, but also helping them understand why they matter.
Speaker 1:Yes. Give them the why.
Speaker 2:Yes. It's not just about memorizing a formula and being able to pick out an outlier on a graph. Right. It's about equipping them to think critically about data, to be able to interpret those outliers and understand their potential impact.
Speaker 1:I love that. It Because data's everywhere.
Speaker 2:Of where?
Speaker 1:And it's only becoming more prevalent in our world.
Speaker 2:Absolutely.
Speaker 1:So giving our students these skills, it's not just about acing algebra class.
Speaker 2:Right.
Speaker 1:It's about being an informed citizen, being able to make sense of the world around them.
Speaker 2:Exactly. And this lesson really lays the groundwork for that.
Speaker 1:It does. Okay. So we're nearing the end of this lesson plan. What does it look like to wrap up this topic with students?
Speaker 2:Well, the illustrative mathematics team provides some really great suggestions in the teacher notes.
Speaker 1:Oh, good.
Speaker 2:One idea they have is to bring it back to that real world context. So after students have explored outliers with the health care spending data, maybe have them think about other areas where outliers might pop up.
Speaker 1:Oh, I like that. So it could be like a brainstorming session.
Speaker 2:Exactly. Get them thinking about outliers in sports, in entertainment, in their own lives.
Speaker 1:It could even be something like, what is an outlier in the context of our school?
Speaker 2:Oh, that's a great idea. And as they're brainstorming, encourage them to think about those same questions we've been discussing. How do we identify outliers? What impact might they have? And most importantly, what might they tell us about the data?
Speaker 1:Right. Because sometimes outliers are just flukes, but other times, they point to something really significant.
Speaker 2:Exactly. And that's what makes them so fascinating.
Speaker 1:Now I'm looking ahead at the next activity in this lesson, and it's called origins of outliers.
Speaker 2:Oh, yeah. This one is interesting. It takes that whole idea of context and really digs in.
Speaker 1:Okay. I'm intrigued. Tell me more. Okay. So origins of outliers.
Speaker 1:This sounds like we're really getting into the nitty gritty here.
Speaker 2:We are. And they start off, again, with these really relatable examples.
Speaker 1:Perfect. Because at this point, I need relatable.
Speaker 2:So they talk about things like, you know, if you were to look at the average income
Speaker 1:Alright.
Speaker 2:Of people in a certain town
Speaker 1:Yeah. Okay.
Speaker 2:And you have someone like Bill Bates who lives there.
Speaker 1:Oh, yeah. That's gonna skew things a bit.
Speaker 2:You know, his income is going to be this huge outlier.
Speaker 1:Totally.
Speaker 2:And so it's making the average income look a lot higher than it really is.
Speaker 1:Right. It's not really representative of what's actually going on.
Speaker 2:Exactly. And then they talk about, like, you know, if you're measuring the heights Okay. Of students in a class
Speaker 1:Okay.
Speaker 2:And you have one student who's, you know, maybe they're, like, a really tall basketball player.
Speaker 1:Right. They're gonna be way up there.
Speaker 2:Yeah. And so it's, again, skewing that data.
Speaker 1:Right. So it's about context again. It's not always about just tossing out the outlier.
Speaker 2:Right. And it's asking those questions. Why is this an outlier? Is it a mistake? Is it just something that's unusual but still valid?
Speaker 1:Exactly.
Speaker 2:And what do we do with that information?
Speaker 1:Okay. This has been so insightful. Any final thoughts for our listeners as they head back to their classrooms?
Speaker 2:I think the biggest thing is just to remember that outliers are more than just these weird numbers that we try to ignore. Right. They can actually tell us a lot
Speaker 1:Yes. They can.
Speaker 2:About our data
Speaker 1:Absolutely.
Speaker 2:And so encourage your students to be curious about them.
Speaker 1:Yes. Curiosity, not fear.
Speaker 2:Exactly.
Speaker 1:I love that.
Speaker 2:Don't be afraid of outliers.
Speaker 1:Well, that is about all the time we have for today. Thank you so much for joining me on this deep dive. I always learn so much from you.
Speaker 2:Oh, it's my pleasure as always.
Speaker 1:And to our listeners, thank you for being here. We'll be back next time with another deep dive into the world of education. Until then, keep those brains buzzing.