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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 this, you're at the grocery store. Right? And you're trying to see if that giant bag of oranges is gonna like fit your budget. Are you really gonna weigh every single orange?
Speaker 1:Probably not. You're gonna like mentally estimate using linear thinking, and that's exactly what we're diving into today with linear models. It's an illustrative mathematics lesson plan, and it's for algebra 1. And just in case you're wondering illustrative mathematics, what even is that? We'll make sure to unpack that too, so no worries.
Speaker 1:Think of this deep dive as like getting a backstage pass. We're not just looking at what we teach with linear models, but like the why behind how this particular lesson is structured why it's so good. And to help us really get into the pedagogy of it all, we have expert name with us today. Are you ready to reverse engineer some learning?
Speaker 2:Absolutely. I'm ready. We're gonna find all those moments that you can bring to your students.
Speaker 1:I love it. So this lesson starts off with an activity called notice and wonder, crowd noise. And it's basically a scatter plot showing, like, how full a stadium is and how that relates to how loud it gets. But here's the thing, They already include a linear model on the graph. So what's the strategy behind that?
Speaker 2:Well, this is where illustrative mathematics really stands out. You see, instead of just throwing formulas at the students right away, they start by asking teachers to ask their students. What do you notice? What do you wonder about the graph? It's all about inquiry based learning, and research actually shows that it really helps students become more curious and gets them thinking more critically before even starting the actual lesson.
Speaker 1:That makes sense. So it's like they're making their own observations before you even tell them what they're supposed to be looking for.
Speaker 2:Exactly. And what I find interesting is that the lesson plan actually mentions some of the fun observations students might have. Like, how the graph shows at a certain point, there are 3 different noise levels all for pretty much the same crowd size. Oh, wow. Really?
Speaker 2:That's actually kinda funny when you think about it. It's, like, 65,000 people can be totally silent or insanely loud all at the same time. Yeah.
Speaker 1:And it's those little things that can lead to some really interesting discussions about outliers or how data can be all over the place. And, ultimately, how linear models, while useful, don't always perfectly reflect what's going on to the real world.
Speaker 2:For sure. Okay. So we've got them thinking. Now it's time to get to the next activity. Orange, you glad we're boxing first?
Speaker 2:I have to say, I'm already loving this title.
Speaker 1:It's a classic, and it uses something that we can all relate to. Like, how much will a box of oranges weigh based on how many oranges are in it? Seems simple enough, but there's a lot to learn from this activity.
Speaker 2:Okay. I'm listening. What's so great about boxing fruit?
Speaker 1:It's about making those abstract math concepts that students always hear about like tangible. The lesson actually suggests showing students a video of oranges being put into a box that's on a scale Yeah. So they can write down the weight for different numbers of oranges and actually plot those points on a graph. Then they try to draw a line that best fits the data. I see what you're getting at.
Speaker 1:So it's not just about putting dots on a graph. It's about actually seeing how the number of oranges and the weight are related in real life. That's a really cool approach.
Speaker 2:It is. And think about all the light bulb moments students will have. They'll realize that the slip of the line isn't just some random number. It's actually telling them the approximate weight of each orange, and the atricep is showing them the weight of the empty box itself. Plus, they're learning that the further away you get from the data you've measured, the less accurate your predictions are gonna be.
Speaker 1:Yeah. So figuring out the weight of a box with 11 oranges is one thing. But using that same line to figure out how much 50 oranges would weigh, that's where it gets a little iffy. Right?
Speaker 2:Yeah. Exactly. And the best part about this lesson is it doesn't try to pretend there aren't any limitations. Like, it actually encourages teachers to ask those what if questions, like, what are some reasons why the weight of 50 oranges in real life might be different from what our model predicts?
Speaker 1:That's such a good question. I mean, 50 oranges might not even fit in the box, or some oranges could just be bigger than others. Right?
Speaker 2:Exactly. It helps students think about the assumptions we're making when we use these linear models in the real world, and it shows them that math isn't always about finding the right answer. Sometimes it's about understanding what we don't know and when our tools might not be perfect.
Speaker 1:Makes sense. So we've got them thinking about slope intercept and the limits of linear models. What comes after that?
Speaker 2:Well, the lesson plan smoothly moves from boxing fruit to something called food markup. In this activity, students get to see the connection between what it costs to make food and what the restaurant actually sells it for. This time, they're given both the scatterplot and the linear equation.
Speaker 1:Oh, interesting. So now instead of building their own model, they're trying to figure out what this one means.
Speaker 2:Exactly. It's about shifting from building something to understanding what it means. Students have to figure out what the slope and y intercept represent in this new scenario. For example, let's say the slope is 3.48. That actually tells you how much the restaurant is marking up the price.
Speaker 2:So for every extra dollar they spend on ingredients, the price of that dish goes up by almost $3.50.
Speaker 1:Now that is a light bulb moment if I've ever heard one. Mhmm. It's not just an equation anymore. It's like a whole business strategy. What a great way to connect those abstract ideas to something relatable.
Speaker 2:Right. And it gets even better. The lesson even suggests that teachers ask students if it makes sense for a dish with 3 ingredients to still have a price based on the y intercept.
Speaker 1:Now that's a thinker. Of course, there are other costs like paying employees and rent. It's not just about the ingredients.
Speaker 2:Exactly. That question really pushes students to think critically and consider other factors that might be affecting things.
Speaker 1:This lesson plan is really smart. So far, they've done boxing fruit. They've explored food pricing. Is there anything else?
Speaker 2:Oh, there's more. The next one is called the slope is the thing, and it throws students right into a bunch of different situations, all represented by scatter plots, like reaction time versus how old you are or how the price of bananas changes depending on how many you buy, all sorts of stuff.
Speaker 1:Like a linear model buffet. I love it.
Speaker 2:Exactly. And for each situation, students really have to put on their thinking caps and work out what the slope and I intercept are telling them. So for example, with the bananas, they might say something like, every time you add another banana, the price goes up by about 44¢.
Speaker 1:So really getting comfortable with understanding what the numbers mean in different situations.
Speaker 2:That's the goal. And it's not even about always getting the right answer. It's about giving students the confidence to use what they've learned in new and different ways.
Speaker 1:That makes sense. Yeah. So to wrap it up, we've covered a lot of ground here. This illustrative mathematics lesson is really well designed. It starts with hands on activities and it gets into more abstract interpretations.
Speaker 1:But let's be real for a second. What are some of the things that students might get tripped up on? Like, what are the common misconceptions that teachers should be aware of?
Speaker 2:Oh, definitely. Even with the best lesson plans, students are gonna run into some tricky parts. One thing that often trips them up with linear models is really understanding the y intercept, especially when you can't see 0 on the graph.
Speaker 1:Yeah. I can see how that would be confusing. It's easy to look at the y intercept and not really think about what it actually means if you're not careful about the scale of the graph.
Speaker 2:Exactly. And the lesson plan actually addresses this right away in that first crowd noise activity. It encourages teachers to ask their students to really consider what a noise level of, like, a 105 decibels really means in that specific situation. Like, is it actually that loud? It's about making those numbers feel real to them.
Speaker 1:It's all about making those connections. So are there any other common misconceptions?
Speaker 2:Another one is interpreting the slope when the scales on the x and y axis are different. Students sometimes have a hard time figuring out the rise overrun, especially if they're just trying to do it visually.
Speaker 1:So it's helpful to have them actually calculate the slope using points on the line.
Speaker 2:Absolutely. And then there's the classic challenge of transferring their knowledge. Like, a student might understand these concepts perfectly fine when it comes to oranges and boxes, but then struggle to use that same logic with something like how temperature and volume are related.
Speaker 1:It's like they know the rules, but they're not sure how to use them in different situations.
Speaker 2:Mhmm.
Speaker 1:So what can teachers do to help with that?
Speaker 2:It really just comes down to giving them lots of different examples and practice opportunities. The more they see these concepts being used in different ways, the more it'll click for them.
Speaker 1:That makes sense. So how does this illustrative mathematics lesson wrap things up?
Speaker 2:Well, it goes right back to the beginning with the stadium noise. The last activity is called roar of the crowd, and it gives students another chance to practice everything they've learned with that same scenario.
Speaker 1:Oh, cool. So it brings the lesson full circle. What do they actually do in this cool down activity?
Speaker 2:This time, they're given the equation for the linear model, and they have to explain what the slope and way intercept mean. And they also have to make predictions about how loud it would be with a certain number of people in the stadium.
Speaker 1:So they're putting all those skills they've been working on to the test one last time. I like how this lesson plan seamlessly blends those hands on parts with more abstract thinking.
Speaker 2:Me too. And it ends with a really interesting question. It suggests that teachers ask students if it makes sense for an empty stadium to have a noise level of 22.7 decibels like the I intercept shows in this case.
Speaker 1:That's a good one. It reminds them that they can't just blindly use math models without considering if they make sense in the real world.
Speaker 2:For sure. It's all about thinking critically and remembering that math isn't just about plugging numbers into formulas. It's about using those tools in smart ways and understanding when they might not be perfect.
Speaker 1:Well, I'm really glad we did this deep dive into linear models today. It's so clear that this illustrative mathematics lesson plan is more than just a bunch of activities. It's designed to really guide students towards understanding why this stuff matters.
Speaker 2:Totally agree.
Speaker 1:A huge thank you to the authors of illustrative mathematics for giving us so much to think about.