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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.)
Ever had one of those moments prepping a lesson, like, on mean, median, and maybe?
Speaker 2:And you just now know. Yeah. Some students are gonna, like, totally zone out
Speaker 1:Right.
Speaker 2:The second you even say data.
Speaker 1:It's like you could practically hear that groan coming. Right?
Speaker 2:Exactly. Like a wave of boring just washing over them. Yep. But what if
Speaker 1:What if we could make it click?
Speaker 2:You know? Yeah.
Speaker 1:Turn that groan into, like, that's what this is all about.
Speaker 2:Right. Exactly.
Speaker 1:So that's our deep dive today, helping you guys, the teachers.
Speaker 2:Make measures of center and variability actually interesting.
Speaker 1:Yes.
Speaker 2:We're looking at excerpts from an algebra teacher's guide
Speaker 1:Okay.
Speaker 2:And some extra notes we've got
Speaker 1:to really bring this topic to life
Speaker 2:Awesome.
Speaker 1:Because it's important, but sometimes it's presented so dry.
Speaker 2:It is. And, you know, it's not just about memorizing formulas and things.
Speaker 1:This stuff is actually really powerful. Like, imagine your students being able to see right through a misleading graph Oh, yes. Or just understanding all those data visualizations everyone and their mother's sharing online these days.
Speaker 2:Okay. I'm already seeing how this connects to that bigger picture of data literacy. Right? It's like we're giving them the superpower to decode the world.
Speaker 1:Yes. Exactly. Now this lesson plan that we're gonna look at Okay. It starts with actually calculating mean, median
Speaker 2:Okay.
Speaker 1:M a d and IQR.
Speaker 2:Got it.
Speaker 1:And it builds on what they should already know about data analysis.
Speaker 2:So it kinda sets the stage for, like, really going deeper into interpreting data later.
Speaker 1:Oh, I see. So this is like laying the foundation
Speaker 2:Exactly.
Speaker 1:For them to become these, like, data detectives.
Speaker 2:Right.
Speaker 1:But before we even get there, can we just do a quick, like, level set
Speaker 2:Of course. Yeah.
Speaker 1:On what these terms actually mean?
Speaker 2:Yeah. Absolutely.
Speaker 1:Because I feel like even as adults, we sometimes forget.
Speaker 2:Right. For sure.
Speaker 1:So you've got your measures of center, like the mean
Speaker 2:Okay. Which is
Speaker 1:just fancy word for average. Right?
Speaker 2:Right. Exactly.
Speaker 1:Adding them all up, dividing by how many there are
Speaker 2:Yep.
Speaker 1:And then you've got the median
Speaker 2:Yes.
Speaker 1:Which is the middle value when you line them all up, which is really handy when you have those outliers.
Speaker 2:Yes. Especially then.
Speaker 1:Trying to mess everything up.
Speaker 2:Yes. Definitely.
Speaker 1:That one kid who eats, like, 10 slices of pizza Right. When everyone else is having 2, and it completely skews the average.
Speaker 2:You got it. And that's where measures of variability are really important because they give us a sense of the spread Okay. Of how clustered or not clustered the data is.
Speaker 1:Okay.
Speaker 2:So you've got things like the interquartile range or IQR. Right. And that tells us about the middle 50% of the data.
Speaker 1:Got it.
Speaker 2:And then there's mean absolute deviation or MAD, which measures the average distance of each data point from the mean.
Speaker 1:So it's like IQR is seeing how tightly packed the middle of the crowd is at a concert.
Speaker 2:Okay. Yeah.
Speaker 1:And Mei D is, like, measuring how far everyone wandered from their friends during the show.
Speaker 2:That is a fantastic way to put it.
Speaker 1:Okay. I like it.
Speaker 2:So this lesson plan we're looking at
Speaker 1:Okay.
Speaker 2:It starts off with this, I think, brilliant warm up activity.
Speaker 1:Using true, false statements about mean and median. Oh. Right. So it's like sneaking in those common errors students make.
Speaker 2:Okay.
Speaker 1:And getting them to really think critically about it.
Speaker 2:Oh, I love that. Like, what kind of statement are we talking about?
Speaker 1:So one of them says the median of the dataset is 9 for the data. 4, 5, 9, 110.
Speaker 2:Okay.
Speaker 1:But here's the thing. They're not in order.
Speaker 2:Yes. Classic mistake.
Speaker 1:Right.
Speaker 2:They see a 9 in the middle Exactly. And they think, boom, median.
Speaker 1:Exactly.
Speaker 2:But they forgot that crucial first step Yes. Puts them in order.
Speaker 1:It makes all the difference.
Speaker 2:It does.
Speaker 1:And it highlights how important it is to actually understand what the median represents
Speaker 2:Yes.
Speaker 1:Not just blindly apply some formula.
Speaker 2:Well, 100%.
Speaker 1:Right. Because if you don't understand what it represents
Speaker 2:You're just gonna make those mistakes over over
Speaker 1:again. Exactly. It's not gonna stick. Exactly. Now speaking of making it stick
Speaker 2:This lesson plan doesn't just throw formulas at students.
Speaker 1:Right.
Speaker 2:It brings the concepts to life with activities like heartbeats.
Speaker 1:Heartbeats.
Speaker 2:Yeah.
Speaker 1:Okay. Tell me more about that.
Speaker 2:So it uses real heart rate data
Speaker 1:Okay.
Speaker 2:To make calculating the median and IQR.
Speaker 1:More engaging and relatable for students.
Speaker 2:Okay. So they're using real data. Yes. That's awesome.
Speaker 1:And the lesson plan even has this note about how those fitness trackers and smartwatches
Speaker 2:You know how they actually collect the heart rate data? Oh, geez. So it connects it to the technology that the kids are using.
Speaker 1:I love that.
Speaker 2:For people to
Speaker 1:That real world connection. Right?
Speaker 2:Sick. It's
Speaker 1:not just some abstract concept in a textbook.
Speaker 2:It's about them.
Speaker 1:This is real stuff.
Speaker 2:Exactly.
Speaker 1:Okay. I like it.
Speaker 2:And it gives you some sample data to use, like 72, 75, 81 Okay. 76, 76, 77, 79, 78.
Speaker 1:God. So we've got, like, a list of heart rates.
Speaker 2:Yes. Exactly for them to practice with.
Speaker 1:Okay.
Speaker 2:And there's even this optional graphic organizer.
Speaker 1:Oh, I love a good graphic organizer.
Speaker 2:To help them visualize that IQR calculation.
Speaker 1:Okay.
Speaker 2:Which, as you know, some students really need that visual.
Speaker 1:Huge. Especially with this age group.
Speaker 2:Yeah. They might be more visual learners.
Speaker 1:Okay. So walk me through it.
Speaker 2:Okay. So first
Speaker 1:How would we find the IQR with this heart rate data?
Speaker 2:So we need the median. Right.
Speaker 1:Right.
Speaker 2:Which, remember, it's the middle value
Speaker 1:Nice.
Speaker 2:But we have an even number of data points.
Speaker 1:So we gotta find the average of those two middle value
Speaker 2:Yeah. In between.
Speaker 1:Which, in this case, are 76 and 77.
Speaker 2:Okay.
Speaker 1:So our median heart rate is 76.5 beats per minute. 76.5. Got it. Alright. And then for the IQR, we're figuring out the quartiles.
Speaker 2:Yes. You're on a roll.
Speaker 1:Okay. So, like, q one and q three.
Speaker 2:Exactly.
Speaker 1:What are those again?
Speaker 2:So q one is the median of the lower half of the data Okay. And q three is the median of the upper half.
Speaker 1:Got it.
Speaker 2:So looking at our heart rates, q one would be the median of 72, 75, 76, and 76 Okay. Which is 75.5.
Speaker 1:75.5. Okay. The q three
Speaker 2:The median of the upper half. So 77, 78, 79, 81.
Speaker 1:Which, if you work it out, is 78.5. 78.5. Alright.
Speaker 2:So we've got our quartiles.
Speaker 1:Okay.
Speaker 2:What's the last step to get that IQR?
Speaker 1:Alright. So we subtract q one from q three.
Speaker 2:Okay.
Speaker 1:Simple as that. So 78.5 minus 75.5.
Speaker 2:Okay.
Speaker 1:This gives us an IQR of 3.3. So that's 3 beats per minute in this case?
Speaker 2:Beats per minute. Exactly.
Speaker 1:Got it. So the middle 50%.
Speaker 2:Of those heart rates fall within that 3 BPM range.
Speaker 1:You got it.
Speaker 2:Okay. It's cool to see it all come together.
Speaker 1:Right. And the lesson plan doesn't stop there.
Speaker 2:Okay.
Speaker 1:It's got heartbeats part 2.
Speaker 2:Which tackles a midi Okay. In a really cool hands on way.
Speaker 1:Okay. I like where this is going.
Speaker 2:Using pennies and a meter stick.
Speaker 1:Pennies. Right.
Speaker 2:Because I know. Right?
Speaker 1:I'm intrigued. Tell me more about this penny powered math.
Speaker 2:So it takes my d, which can be kind of abstract.
Speaker 1:Yeah.
Speaker 2:Right? Thinking about the average distance from the mean.
Speaker 1:And it makes it visual, tangible. You can touch it.
Speaker 2:Okay.
Speaker 1:And the lesson even tells you exactly where to put each penny on the meter stick.
Speaker 2:To represent the different data points.
Speaker 1:Oh, wow. So they give you, like, a template to follow?
Speaker 2:Yeah. Exactly.
Speaker 1:That's helpful.
Speaker 2:So you can see it, the spread of the data, how far each point is from the mean Okay. Which would be the middle of the meter stick. Right? Right. So smart.
Speaker 2:It is. It's a You could actually see it exactly.
Speaker 1:Instead of it just being numbers on a page.
Speaker 2:And you know what I love about this activity?
Speaker 1:What's that?
Speaker 2:It really lets them explore. Right?
Speaker 1:Okay.
Speaker 2:Like, what happens if we move this penny over here?
Speaker 1:Yes. Yes.
Speaker 2:Or what if we add a new penny way out here?
Speaker 1:What happens to the MAD then? Exactly. Oh, that's so good.
Speaker 2:It becomes this dynamic thing where they can make predictions
Speaker 1:Yes.
Speaker 2:Test them out, really understand.
Speaker 1:I love that they're like little scientists.
Speaker 2:Exactly.
Speaker 1:But even with the best activities, the most engaging activities Right. Some students might still hit some roadblocks.
Speaker 2:What are some of those, like, common sticking points?
Speaker 1:So the lesson plan mentions.
Speaker 2:That calculating the median with an even number of data points
Speaker 1:Right.
Speaker 2:That can be tricky.
Speaker 1:Yes. I can see
Speaker 2:students might forget that step.
Speaker 1:Where you have to find the average
Speaker 2:Right. Right.
Speaker 1:Of those two middle values.
Speaker 2:Not just pick 1. You gotta go in between.
Speaker 1:Exactly. You gotta meet in the middle.
Speaker 2:Okay. And what about map
Speaker 1:b? I feel like absolute value
Speaker 2:Yes.
Speaker 1:Always trips students up. Always. It's one of those things.
Speaker 2:It's like, why is it always positive?
Speaker 1:Right. Distance can't be negative.
Speaker 2:Exactly.
Speaker 1:But it's easy to forget.
Speaker 2:It is. And that's where a visual, like a number line
Speaker 1:Oh, yeah. I love a good number line.
Speaker 2:Can be so helpful.
Speaker 1:Visuals are key.
Speaker 2:Yes. Absolutely.
Speaker 1:Because they can actually, like, count the spaces
Speaker 2:Exactly.
Speaker 1:And see the distance.
Speaker 2:And see that it's always positive no matter what direction you're going.
Speaker 1:Okay. I like that.
Speaker 2:Because distance is always positive in real life.
Speaker 1:Exactly.
Speaker 2:The notes also mentioned having students work backward
Speaker 1:Oh, yes.
Speaker 2:Where they create their own datasets.
Speaker 1:With, like, a specific Mady or IQR in mind.
Speaker 2:I love that approach.
Speaker 1:Have you tried that?
Speaker 2:Oh, it's a game changer. Really? It forces them to think differently about it.
Speaker 1:Not just plugging in numbers, but really understanding how the data affects those measures.
Speaker 2:Oh, I see. So they have to, like, think strategically
Speaker 1:Yes. Exactly.
Speaker 2:To make the data fit.
Speaker 1:It's almost like they become these data artists
Speaker 2:Okay.
Speaker 1:Creating their own masterpieces.
Speaker 2:I love that.
Speaker 1:And that's what I love about all this.
Speaker 2:It's not just teaching them to calculate something.
Speaker 1:Right.
Speaker 2:It's about data literacy.
Speaker 1:Right.
Speaker 2:Being able to think critically about data
Speaker 1:Essential skills.
Speaker 2:In the real world, which is what we want.
Speaker 1:A 100%.
Speaker 2:Now the lesson plan also mentioned
Speaker 1:Creating a classroom display.
Speaker 2:Oh, yes.
Speaker 1:Showcasing all this, the measures of center and variability.
Speaker 2:I love that idea.
Speaker 1:Building it out as you go through the unit.
Speaker 2:So it's like a constant reminder
Speaker 1:Yes.
Speaker 2:Of what they're learning.
Speaker 1:And it grows as their understanding grows.
Speaker 2:Exactly.
Speaker 1:Okay. I love it.
Speaker 2:So as we wrap up this deep dive is,
Speaker 1:like, one key takeaway.
Speaker 2:You hope teachers will, you know, take with them.
Speaker 1:We talked about some specific activities.
Speaker 2:Right? Heart rates, pennies?
Speaker 1:Yes. But don't be afraid to get creative.
Speaker 2:Okay.
Speaker 1:Think about your students.
Speaker 2:What kind of data would really resonate with them?
Speaker 1:Make it relevant.
Speaker 2:Exactly.
Speaker 1:Well, that's such a good point.
Speaker 2:Because if they're interested, they're gonna be engaged.
Speaker 1:Exactly. That's the key.
Speaker 2:Awesome. Well, this has been fantastic.
Speaker 1:Thank you.
Speaker 2:Thank you so much for diving deep with us.
Speaker 1:My pleasure.
Speaker 2:And to our listeners, keep exploring this stuff.
Speaker 1:Find what works for you, for your students Absolutely. And keep those moments coming.
Speaker 2:Yes. That's what it's all about.
Speaker 1:A huge thank you to the authors of Illustrative Math
Speaker 2:for this awesome lesson plan.
Speaker 1:And to our listeners, keep up the great work.
Speaker 2:We'll catch you next time.