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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 imagine, you're dying to get your hands on this new video game.
Speaker 2:Who isn't?
Speaker 1:Exactly. And you see it at, like, 2 different stores. Store a has it marked down 20%. Store b's got a huge 35% off sticker.
Speaker 2:Sounds like a no brainer. Right?
Speaker 1:Right. You'd grab it from store b, do a little victory dance. But hold up a sec. What if store a, their original price was $40, and store b, it was 60?
Speaker 2:Percentages, they can be deceiving.
Speaker 1:Sneaky little.
Speaker 2:That's where our deep dive today comes in handy.
Speaker 1:I was gonna say without context, percentages don't tell us the whole story.
Speaker 2:They don't. Context is everything, and that's what relative frequency tables are all about, giving us that bigger picture.
Speaker 1:Okay. So we're looking at, an illustrative math lesson plan, algebra 1. Right?
Speaker 2:Right. And you're the teacher at this time. Gotta bring this concept to life for your students.
Speaker 1:And I want those moments, like help them become data detectives, not just number crunchers. So what's the big idea here? Relative frequency tables. Break it down for us.
Speaker 2:So these tables, what they do is they take those raw numbers, like our video game prices, and they turn them into, like, proportions within categories.
Speaker 1:Okay. So it's not just, like, how many games are on sale, but what that discount actually means, right Yeah. Compared to the original price.
Speaker 2:Yeah. Or even within the context of different stores. Right? Like we saw So
Speaker 1:it's like adding context to the numbers, giving them a backstory, making them more human.
Speaker 2:Exactly.
Speaker 1:Yeah.
Speaker 2:And this lesson plan, it digs into 3 main ways to do this, overall, row, and column relative frequencies. Think of them like 3 different lenses on a camera.
Speaker 1:Oh, I like that.
Speaker 2:Same dataset, different perspectives.
Speaker 1:Alright. I'm ready to zoom in with those lenses. What's our first activity?
Speaker 2:So this one uses a table about teachers and their degrees. Pretty straightforward. Starts with a simple question. Does having a higher degree make you more likely to be a teacher? Interesting.
Speaker 2:At first glance, just looking at the raw numbers, you might think, yeah, maybe so. You know, you might see more teachers with master's degrees, for example, in the data.
Speaker 1:Right. Right. But more teachers having higher degree, it doesn't automatically equal a higher likelihood of teachers having them, does it?
Speaker 2:Exactly. And the lesson, it sets up this kind of, like, apparent contradiction. It gets the students wrestling with that.
Speaker 1:Oh, I see where this is going.
Speaker 2:It doesn't even calculate the percentages in this first activity.
Speaker 1:Keeps them in suspense.
Speaker 2:Yeah. But it gets them thinking, okay. We need a different way of looking at this. Like, setting up a puzzle, they know something isn't quite adding up yet.
Speaker 1:Alright. So the suspense is killing me. What's next?
Speaker 2:Alright. Next up, we've got cats and dogs, the city versus the country.
Speaker 1:200 people surveyed.
Speaker 2:200 people. Yep. And this one, this one uses 4 different tables to really hammer home this point.
Speaker 1:Oh, very clue board game.
Speaker 2:It is. Each table adds another layer. It's like piecing together the clues. Right?
Speaker 1:I hope no cats were harmed in the making of this data.
Speaker 2:No cats were harmed, but we're definitely analyzing their preferences.
Speaker 1:Alright. So we start with, like, just the raw numbers, how many people fit each description.
Speaker 2:Right. Then we see the percentages of the total group.
Speaker 1:Okay. So far so good. But that's just setting the scene. Right? Where's the moment?
Speaker 2:Table 3. That's where it clicks. That's where those column relative frequencies come in.
Speaker 1:Oh, I see.
Speaker 2:They see how comparing the columns can reveal if, say, city dwellers really are more into cats than their country counterparts.
Speaker 1:So if I'm explaining this in class, I might say, imagine you're a pet food company. Right? You're trying to figure out where to put a billboard.
Speaker 2:Perfect example.
Speaker 1:Do you target the city folks or the country folks? Which group is more likely to be swayed by, like, a picture of a cute cat?
Speaker 2:And boom, there's your column frequency relevance.
Speaker 1:And then we flip the script with table 4. Right? Yeah. Row relative frequencies. This is like saying, okay.
Speaker 1:We've seen how city versus country folks feel about cats versus dogs. But now let's compare the cat lovers to the dog lovers within each location.
Speaker 2:Exactly. Like, are city cat people more intense in their love than country cat people?
Speaker 1:Oh, the plot thickens. Yeah. I'm already feeling like a data detective.
Speaker 2:Yeah.
Speaker 1:Okay. Ready for the next case?
Speaker 2:Alright. Get your magnifying glass. Activity 3 throws us right into a medical mystery.
Speaker 1:I love a good medical mystery.
Speaker 2:The age old question, does vitamin c actually help with colds? Oh. And at first, the data is, well, it's messy.
Speaker 1:So it's a real head scratcher. I'm already picturing myself in a lab coat analyzing data under a microscope.
Speaker 2:Maybe hold off on that lab coat for now, but you will be analyzing data. The lesson has students build 3 tables here.
Speaker 1:Three tables. Okay.
Speaker 2:First one, it it uses overall relative frequencies, which is kinda like, you know, you get a basic snapshot.
Speaker 1:A very blurry snapshot.
Speaker 2:Yeah. Gives you an idea, but not enough to, like, diagnose anything for sure.
Speaker 1:I need to zoom in, get clearer picture
Speaker 2:Exactly. Enhance. That's where our good friend, call them relative frequencies, comes in again.
Speaker 1:Ah, I knew they'd be back.
Speaker 2:Table 2, that's where we really isolate the effect of each treatment. Those who took vitamin c versus those who, you know, just got the sugar poll.
Speaker 1:The placebo group. Right.
Speaker 2:Yeah. Yeah.
Speaker 1:So we're comparing apples to apples within each column. Right. Did people who took vitamin c get better at a higher rate than those who didn't, regardless of any other factors?
Speaker 2:You got it. And then because, you know, we love a good plot twist
Speaker 1:Of course.
Speaker 2:Table 3, we flip the perspective again. Row relative frequencies.
Speaker 1:Okay. My brain's doing backflips now. I gotta keep up. So with the row frequency, we're answering a different question. Right?
Speaker 2:We are.
Speaker 1:It's like, if you took vitamin c, were you more likely to be in the got better group than if you took the placebo?
Speaker 2:You're getting it, and that's the beauty of relative frequency. It all comes down to what question you're asking.
Speaker 1:So you really have to think critically. This is more than just plugging numbers into a formula. It's about interpretation.
Speaker 2:Absolutely. Yeah. Digging deeper. And speaking of things that might trip students up Oh,
Speaker 1:there are always things that trip them up. Percentages can be tricky.
Speaker 2:They can. They can. The source material, it points out how students might mix up their totals using the wrong denominator for calculations. You know, like
Speaker 1:Classic mistake.
Speaker 2:Thinking there are a 100 animals at the zoo just because 50% are monkeys?
Speaker 1:Oh, no. Poor math skills.
Speaker 2:You need to know how many animals there are total, not just the monkey count.
Speaker 1:Right. Gotta start with that solid foundation. What other common pitfalls do they point out?
Speaker 2:Misinterpreting column and row percentages.
Speaker 1:Oh, like trying to add them up when they represent parts of different whole.
Speaker 2:Exactly. Classic mistake. 50% of the class loves pizza, 60% loves ice cream, therefore, 110% must love both.
Speaker 1:Wait. That doesn't add up. Literally. It's like those percentages are slices from different pies, not pieces of the same one.
Speaker 2:You got it.
Speaker 1:Okay. So those are some common calculation errors. Did the lesson plan say anything about students struggling with the why behind all of this? Like, why is relative frequency so important in the first place?
Speaker 2:They do. And it's a good point. Students can get so caught up in the, you know, the mechanics of it all, they kinda miss the bigger picture.
Speaker 1:Right. It's like learning all the dance moves but not understanding the music.
Speaker 2:Exactly.
Speaker 1:So how do we help them find the music, so to speak? How do we make relative frequency really click for them?
Speaker 2:Well, that's where we circle back. Right? We bring it back to those initial examples, the video games, the cats and dogs, the teachers.
Speaker 1:Show them the real world connections.
Speaker 2:Yeah. Show them how those relative frequencies can reveal the patterns that those raw numbers might obscure.
Speaker 1:It's like giving them X-ray vision for statistics. They can see beneath the surface.
Speaker 2:And to make sure that, you know, that vision really sticks with them, this lesson wraps up with what they call a cool down, a little writing sample.
Speaker 1:A cool down. I like it. I'm always looking for creative ways to assess my students without stressing them out.
Speaker 2:This one's great. It presents a scenario. Right? Students' handwriting, are they left handed or right handed?
Speaker 1:Okay. Relatable.
Speaker 2:But instead of making them calculate the percentages themselves, this activity gives them a partially completed relative frequency table and asks them to fill in the blanks.
Speaker 1:Oh, that's clever. It's like giving them a puzzle where some of the pieces are already in place, and they have to figure out where the rest go. And in doing so, they're demonstrating their understanding of how those proportions work.
Speaker 2:Exactly. It tests their understanding without, you know, making them do the whole calculation from scratch.
Speaker 1:Love it. So they're filling in the blanks. Is that it, or is there, like, a a final challenge?
Speaker 2:Oh, there's a final challenge. Of course, there is. It asks them to use the table they just filled out to answer a specific question, involves calculating a percentage, but, like, strategically, you know.
Speaker 1:So it brings everything full circle.
Speaker 2:Mhmm.
Speaker 1:They're not just creating these tables in a vacuum. They're using them to solve real problems.
Speaker 2:That's fast.
Speaker 1:Telling them this whole lesson plan is giving me serious teacher envy. It's so well structured. Yeah. I could practically hear my students having those light bulb moments already.
Speaker 2:And to really make those light bulb moments stick the landing, this lesson, it doesn't just stop at the student activities. It also challenges you, the teacher, to leave them with a thought provoking question.
Speaker 1:Oh, cliffhanger. Gotta love a good cliffhanger.
Speaker 2:Right? And this one's all about bringing it back to those misleading percentages we talked about at the very beginning. So, like,
Speaker 1:create our own real world scenario that seems counterintuitive at first glance.
Speaker 2:Exactly. Something using your students' interest. Something that'll make them go, wait, what? And actually wanna dig deeper to find the truth.
Speaker 1:Okay. I think I'm getting some ideas. Like, let's say I I know my students are obsessed with this one video game. Right?
Speaker 2:Okay.
Speaker 1:I could totally create a scenario comparing, like, the price of in game items or something using percentages that seem all messed up at first until you really break it down using what we've learned today about relative frequencies.
Speaker 2:Perfect example. Or let's say, they're all about collecting those, like, limited edition sneakers.
Speaker 1:Oh, yeah. They're always talking about sneakers.
Speaker 2:You can have them analyze some online resale prices, throw in some head scratching discounts, you know, really make them think.
Speaker 1:This is brilliant. It's about taking something they care about, something that feels relevant to their world, and using it to unlock a deeper understanding of these math concepts.
Speaker 2:And who knows? Maybe they'll even start spotting those misleading percentages out in the wild, become savvy shoppers
Speaker 1:Critical thinkers.
Speaker 2:Data detectives, all thanks to a little deep dive into relative frequency tables.
Speaker 1:Yes. Huge shout out to Illustrative Math for always bringing those insightful and engaging math lessons.
Speaker 2:Couldn't agree more. They knocked it out of the park with this one.
Speaker 1:Absolutely. And to all of you listening, thanks for joining us on this deep dive. Remember, it's not just about the numbers, it's about the stories they tell and the questions they inspire. Until next time, keep exploring, keep questioning, and keep on diving deep.