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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.)
Okay. So have you ever had that feeling right before you're about to teach standard deviation, you know, where you can practically see the blank stairs and feel the walls going up?
Speaker 2:Oh, absolutely.
Speaker 1:Like, it's this big scary formula that just sucks all the air out of room, and everyone's just like, ugh, not this again. But, like, what if we could change that?
Speaker 2:Right.
Speaker 1:What if we can make standard deviation, dare I say it, engaging?
Speaker 2:Go out on that fun.
Speaker 1:Okay. Not fun fun.
Speaker 2:Yeah.
Speaker 1:But at least less of a drag.
Speaker 2:You know? Definitely. And that's actually what we're diving into today.
Speaker 1:Yes. So we're taking a deep dive into lesson 12 from illustrative math's algebra 1 curriculum. And it's all about helping our students really get standard deviation, not just fear it.
Speaker 2:And what I like about this lesson plan in particular is it doesn't treat standard deviation as this, like, isolated terrifying concept.
Speaker 1:Right. Like, they're not just throwing a formula at students and be like, here you go. Good luck.
Speaker 2:Right.
Speaker 1:It really smartly connects standard deviation to those measures of variability that they've already encountered.
Speaker 2:Exactly.
Speaker 1:Things like MAD and IQR, which, let's be honest, can be a mouthful on their own.
Speaker 2:Right.
Speaker 1:But they've at least had some exposure to those.
Speaker 2:Yes. And that's something that illustrative math does really well. They really build on that prior knowledge, which I think is so key because, you know, you can tap into something that they already kind of understand. Mhmm. Suddenly, standard deviation just seems a little less scary, a little less like a cryptic code.
Speaker 1:It's less foreign.
Speaker 2:Exactly.
Speaker 1:Because if you can relate it to something, it's like, oh, I've seen something like this before.
Speaker 2:It's like meeting them where they are. It's like, hey. Remember that thing you learned? Like, let's build on that.
Speaker 1:Yes. And speaking of building, this lesson plan has a really smart progression of activities.
Speaker 2:It does.
Speaker 1:It really seems like the writers put a lot of thought into how to scaffold the understanding for students.
Speaker 2:Yeah. Absolutely.
Speaker 1:And one of the things that really stuck out to me was the emphasis on using technology. Oh, yes. Tell me more about that.
Speaker 2:Well, they specifically suggest using Joe Jebra for some of the activities, which is just fantastic for visualizing how changes in data affect the standard deviation.
Speaker 1:Okay. So instead of getting bogged down by the calculation, they can actually see what's happening.
Speaker 2:Exactly. And I think that's really powerful for students because, you know, you can imagine having them create a dot plot of, let's say, their quiz scores or something Mhmm. And then have them drag a single point further away from the mean and then see how that standard deviation changes in real time. Right. Talk about making that abstract formula come alive.
Speaker 1:Oh, I love that. Because it's one thing to memorize a formula. It's a whole other level of understanding to actually visually manipulate the data and see how it impacts that number.
Speaker 2:Right. Oh, that's what that number actually represents.
Speaker 1:Like, it clicks. And they can see, like, oh, it's not just some random number. It's actually reflecting, like, how spread out those points are.
Speaker 2:Right. And it doesn't just stop at visualizing it either. Mhmm. The lesson actually takes it a step further with activities like investigating variability
Speaker 1:Mhmm. Where Oh, yeah. Yeah. Tell me about that one. That one looks really interesting.
Speaker 2:So in this activity, students are, again, like, kind of playing around with data sets. Yeah. But this time, they're looking at how changes affect both the standard deviation and the IQR.
Speaker 1:Okay.
Speaker 2:So they're seeing those two measures side by side.
Speaker 1:So they can compare.
Speaker 2:Yeah. Exactly. So, for example, they might add an outlier to the dataset Mhmm. And see that, oh, wow. It dramatically increases the standard deviation.
Speaker 1:Right.
Speaker 2:But it might have a much smaller impact on the IQR.
Speaker 1:Which makes sense because the IQR is kinda like that middle 50%.
Speaker 2:The outliers don't faze it as much.
Speaker 1:Yeah. Exactly. Exactly.
Speaker 2:And so by having students kinda, like, actively explore those differences, it gives them a much more intuitive understanding of what each of those measures is actually telling us.
Speaker 1:It's like, yeah, what story is it telling? And then can you use that to, you know, compare it? It's like you're giving them, like, data detective tools almost. Yeah. To be like, okay.
Speaker 1:The standard deviation is high here. What does that mean?
Speaker 2:Exactly. What might be going on with this data? Right? Like, moving beyond just calculating the number and actually interpreting what that number means in a context.
Speaker 1:Which is, let's be honest, the whole point of learning statistics in the first place Well, this so we can make sense of the world. Right?
Speaker 2:Absolutely. Now all of this exploration visualization is fantastic, right
Speaker 1:Right.
Speaker 2:For building that conceptual foundation.
Speaker 1:Mhmm.
Speaker 2:But, eventually, we do kind of have to address the elephant of the room, which is that actual formula for standard deviation.
Speaker 1:Right. Because as much as we love the technology and the visuals, at some point, you do have to know how to, like, you know, crunch the numbers a little bit.
Speaker 2:Right. And that's where I think this lesson plan really shines is that it's only after students have had that ample time to play and explore and develop that intuitive sense of standard deviation that they actually introduce the formula, which they do in their are you ready for more section.
Speaker 1:I like that.
Speaker 2:Yeah. So they're not hit with that wall of symbols right off the bat.
Speaker 1:They have that foundation to build on.
Speaker 2:Exactly. Yeah. And they break it down really nicely. They break down the formula step by step, connecting each part of the formula back to those concepts that they've been exploring. Oh.
Speaker 2:So it's not just this magical equation anymore. It's like, oh, this makes sense.
Speaker 1:Right. It's a process like we've been doing this whole time.
Speaker 2:Exactly. Okay.
Speaker 1:So we've got these engaging activities. We're building conceptual understanding, and then we're strategically introducing the formula. But let's be real. Even with the best laid plans, there's always potential pitfalls. Right?
Speaker 2:Oh, for sure. Yeah. And the lesson plan writers actually acknowledge that.
Speaker 1:Yeah.
Speaker 2:They even flagged some common student misconceptions to watch out for, which I think is really helpful.
Speaker 1:Yeah. It's like they've been in our classrooms. Like, they know.
Speaker 2:They know. Here's what to watch out for, teachers.
Speaker 1:Exactly. This is what's gonna happen. So what are some of the big ones that they highlight?
Speaker 2:Well, one that comes up all the time is confusing population standard deviation versus sample standard deviation.
Speaker 1:Yes. The age old, is it the whole group or just part of the group?
Speaker 2:Exactly. And it's understandable, right, especially if students have maybe encountered both of those in previous courses or something.
Speaker 1:Yeah. They've seen it both.
Speaker 2:It's easy to get them mixed up.
Speaker 1:The notation's so similar too. It's really just like one little symbol.
Speaker 2:Right. And this lesson really focuses specifically on population standard deviation Yeah. Which, you know, in algebra 1, they're usually dealing with, you know, entire datasets anyway.
Speaker 1:Right. Right.
Speaker 2:So it makes sense in that context.
Speaker 1:They're not necessarily getting into, like, inferential statistics quite yet.
Speaker 2:Exactly. Yeah. Right. But still, it's something that, you know, as teachers, we wanna be super clear about.
Speaker 1:Oh, absolutely. Because if we're a little fuzzy on it, they're gonna be fuzzy on it.
Speaker 2:Oh, a 100%.
Speaker 1:Think they can sense that. You know?
Speaker 2:And this is where being able to articulate the why is just so important. Like, it's not enough to just say, okay. Use this formula for population and use this formula for sample.
Speaker 1:Right.
Speaker 2:You really need to understand the reasoning behind it.
Speaker 1:Like, why does it matter? Yeah. Because if they get the why, then the how is so much easier to remember.
Speaker 2:Exactly. So, you know, one thing you could do is, like, bring it to life with an example. Right?
Speaker 1:Mhmm.
Speaker 2:Imagine showing them a dataset representing, like, their entire school's test scores versus just, like, a sample of 1 classroom's test scores.
Speaker 1:I like where you're going with this. Yeah.
Speaker 2:And have them calculate both the population and sample standard deviation.
Speaker 1:Because then they'll actually see, oh, it's a little bit larger when it's just our classroom.
Speaker 2:Exactly.
Speaker 1:Because there's naturally gonna be, like, more variability.
Speaker 2:Right. And it kinda opens up a whole discussion about how smaller groups tend to have more variability anyway.
Speaker 1:That's a good point. Yeah.
Speaker 2:Right. It's a subtle difference, but it helps them grasp that, like, we're not just choosing these formulas at random.
Speaker 1:Right. It's not arbitrary.
Speaker 2:It depends on the data we're working with. Yeah.
Speaker 1:Yes. Okay. So we've talked about the activities. We've talked about some of those, like, tricky misconceptions to keep an eye out for. What about wrapping things up?
Speaker 1:Like, how does this lesson suggest kind of, like, bringing it all together at the end?
Speaker 2:Well, they have this, like, really clever true or false cool down activity.
Speaker 1:Oh, I love a good true or false.
Speaker 2:You feel good. Right?
Speaker 1:It's such a good way to, like, gauge understanding and to spark a little bit of discussion.
Speaker 2:Yes. And one of the statements they have is the standard deviation of a dataset that's perfectly symmetrical is always 0.
Speaker 1:Oh, that's a good one because it seems intuitive. Right? Because you think symmetrical, you think, like, balanced, everything in its place.
Speaker 2:Right. Like, it should be 0.
Speaker 1:But then again, standard deviation is about the spread, not just the shape. Exactly.
Speaker 2:So
Speaker 1:you can have something that's perfectly symmetrical. But if all those points are really far away from the mean, the standard deviation's gonna be big.
Speaker 2:Exactly. Like, think about, you know, the normal distribute like, the epitome of symmetry. Right? But that standard deviation determines how wide or narrow that bell curve is.
Speaker 1:That is hex
Speaker 2:So it can be perfectly mirrored on both sides Right. But that doesn't automatically mean a standard deviation of 0.
Speaker 1:There still has to be zero variation for to be true.
Speaker 2:Exactly.
Speaker 1:Oh, that's such a good point and such a good reminder that, like, really understanding standard deviation is more nuanced than just, like, oh, I know the formula.
Speaker 2:Yes.
Speaker 1:It's about really getting those concepts, those nuances.
Speaker 2:Absolutely. And you know what I love about these true or false activities? You can just imagine the classroom discussion. Right?
Speaker 1:Oh, yeah.
Speaker 2:Because some kids are gonna be like, well, it's symmetrical. It's gotta be 0. And others are gonna be like, hold on. Hold on.
Speaker 1:But what about the spread?
Speaker 2:Exactly.
Speaker 1:Yeah. It's such a good way to, like, surface those misunderstandings and really get them talking about it.
Speaker 2:Absolutely. Yeah. It's so much better than just, like, here's the formula. Now go calculate standard deviation on this worksheet.
Speaker 1:Right. Like, nobody wants that. No. So big takeaway for me is this lesson plan from illustrative math. It's, like, such a gold mine of strategies for teaching standard deviation in a way that's, like, engaging and meaningful and, dare I say it, maybe even a little bit fun.
Speaker 2:Well, we might still be debating the fun part.
Speaker 1:But Okay. Maybe not fun.
Speaker 2:But it's definitely a step in the right direction.
Speaker 1:It's definitely more fun than, like, a dry textbook.
Speaker 2:Oh, for sure.
Speaker 1:So huge thanks to illustrative math for putting this incredible resource have had? Head over to our website. Find us on social media. Let's keep this conversation going.
Speaker 2:Oh, shit.
Speaker 1:Because we're all in this together. Yes. Alright. Until next time. Happy teaching, everyone.