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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. Ready to take a deep dive into teaching free fall?
Speaker 2:I am. And don't worry. We won't need a parachute for this one.
Speaker 1:Definitely not. We're gonna see how one lesson plan uses a falling rock, of all things, to unlock some of the secrets of quadratic functions. We've got our hands on a teacher's guide for this lesson, and the focus is on building those quadratic equations organically, you know, not just dropping a formula on them. Right.
Speaker 2:Help them build their own understanding.
Speaker 1:Exactly. Now if you're listening to this, you're probably a teacher yourself, and you might even be getting ready to teach this exact lesson. And let's be real, you wanna knock it out of the park. Right?
Speaker 2:Of course. Who doesn't want their students to have those amazing light bulb moments?
Speaker 1:Totally. So think of us as your teaching assistants for this deep dive. We've got your back.
Speaker 2:We're here to help you make this lesson amazing.
Speaker 1:Okay. Before we jump into the nuts and bolts of it all, let's talk big picture for a second. What are the big mathematical takeaways students should walk away with from this lesson?
Speaker 2:Well, the guide does a great job laying out 3 main goals. The first one is all about getting students to really understand what each term in a quadratic expression actually means in the context of a falling object.
Speaker 1:Right. So it's not just an abstract equation. It's telling a story.
Speaker 2:Exactly. It's like the equation becomes a little window into the motion of the rock.
Speaker 1:I love that. Okay. So understanding the meaning behind the math, what's next?
Speaker 2:The second goal is to get students comfortable representing that falling motion in different ways, like using tables, graphs, and equations.
Speaker 1:Ah, so they can see the connections between those different representations.
Speaker 2:Exactly. It's about understanding how they all tell the same story just in different languages, you know.
Speaker 1:I like the different languages of math. Okay. So we've got meaning. We've got multiple representations. What's the 3rd big goal?
Speaker 2:The third one is the big kahuna. It's getting students to actually write their own quadratic functions to model this free fall scenario.
Speaker 1:So it's about empowering them to build the model, not just follow a formula.
Speaker 2:Exactly. It's all about them taking ownership of the math.
Speaker 1:And that's what makes it stick. This lesson is like laying the foundation for so much more.
Speaker 2:Oh, absolutely. This is their springboard to tackling more complex ideas later on in algebra and physics. We're talking projectile motion, zeros of a function, the vertex, and even the domain all within the amazing world of quadratic functions.
Speaker 1:It's like we're building a mathematical skyscraper here, and this lesson is the bedrock. I'm all about learning by doing, so let's get into the actual activities. The lesson kicks off with a warm up called notice and wonder, and it all starts with this intriguing table of numbers, 0, 16, 64, 144, 256, 400. Just looking at them, what do you think might be going through a student's mind?
Speaker 2:Well, the beauty of notice and wonder is that there are no instructions at first. It's all about sparking their curiosity. The guide encourages letting students just loose on these numbers.
Speaker 1:I can already picture them zeroing in on those multiples of 16.
Speaker 2:Exactly. And maybe even making the leap to perfect squares. But the real magic is that there's no pressure to find the right answer right away. It's about observation, spotting patterns, and getting comfortable with not knowing everything immediately.
Speaker 1:It's like setting the stage for that moment when they finally see it.
Speaker 2:Precisely. This activity beautifully emphasizes the value of noticing the underlying structure before we slap an equation on it.
Speaker 1:And speaking of moments, the next activity is where things get very real world. It's called falling from the sky. Picture this. A rock, a 500 foot building, and a camera snapping photos every second as this rock plummets toward Earth.
Speaker 2:I love how visual that is. Yeah. Students get to analyze the data from those photos looking at the time and the distance the rock fallen.
Speaker 1:It's like they're reliving a classic physics experiment, but with a modern twist. And as they dive into that data, they start to uncover this hidden quadratic relationship.
Speaker 2:And that's where those moments start flying. The guide even highlights a student named Jada who makes the connection between those multiples of 16 from the warm up and the distances the rock falls each second.
Speaker 1:That's amazing. Talk about connecting the dots.
Speaker 2:It's a perfect example of how powerful well sequence activities can be. Jada's realization exemplifies the pivotal shift from simply observing patterns to recognizing the mathematical framework beneath them.
Speaker 1:And then comes the equation itself, d UL 16 Forte Safon. But because they've had all this experience with the activity, it feels less like a foreign language and more like a concise way to express what they've already discovered.
Speaker 2:It's like the equation finally makes sense because they've lived through this scenario.
Speaker 1:It's brilliant. Okay. So we've got patterns. We've got a falling rock. What's next?
Speaker 2:Well, the lesson then throws in a bit of history and a dash of Galileo for good measure.
Speaker 1:Okay. You've piqued my interest. Tell me more about this Galileo connection.
Speaker 2:Yes. It's the Galileo and gravity activity. This one is really clever because it brings in some historical context, you know, grounding the concepts in a real person's work. But it also subtly introduces a new layer of complexity to the problem.
Speaker 1:Oh, I love when they layer in complexity like that. What kind of complexity are we talking about?
Speaker 2:So this time, instead of nice whole number seconds, students are asked to evaluate the equation that d hosly 16 10 Town tests in using a fraction of a second, specifically half a second.
Speaker 1:Interesting. So they really have to think about how the equation works, not just plug in whole numbers.
Speaker 2:Exactly. But it gets even better. The activity also has students compare and contrast 2 different tables. 1, Elena's table, tracks the distance the object has fallen from its starting point, you know, like we've been doing.
Speaker 1:Right. It makes sense.
Speaker 2:But then we have Diego's cable, which focuses on the distance remaining to the ground.
Speaker 1:Okay. So two different ways of looking at the same falling rock.
Speaker 2:Precisely. And get this, the guide cleverly sets the building's height at 576 feet for this activity.
Speaker 1:576 feet. That seems awfully specific. Is there something special about that number? It's not just a random height, is it?
Speaker 2:You know it. They didn't just pull that number out of a hat. Remember our trusty 16 top pattern?
Speaker 1:How could I forget? Seems like it pops up everywhere in this lesson.
Speaker 2:Well, 576 is actually 16 times 36, so it subtly reinforces that connection they made earlier. But the truly brilliant part is how this activity drives home the point that the same phenomenon, in this case, a falling object, can be represented by different yet fundamentally related quadratic functions.
Speaker 1:Oh, I see where you're going with this. So we're talking Dia 16 Tawhid versus what would it be? 576 minus 16 Tawhid.
Speaker 2:I got it. Two sides of the same falling object coin.
Speaker 1:It's like flipping the script, but keeping the math rock rock solid.
Speaker 2:Exactly. It highlights the perspective matters Yeah. Even in math. And to really bring it all home, the guide even suggests using a GeoGebra applet, you know, to really visualize things.
Speaker 1:Because who doesn't love a good visual? Right?
Speaker 2:Yeah.
Speaker 1:Seriously, though, why is incorporating something like GeoGebra so powerful in a lesson like this?
Speaker 2:It all comes down to bridging that gap between the abstract world of equations and graphs and the very real image of an object plummeting to Earth. That visual element seeing the function come to life as the rock falls, it just solidifies understanding in a way that equations on a page sometimes miss.
Speaker 1:I can definitely see that. It makes the math come alive. We've got the patterns, the equations, a little Galileo thrown in for good measure. This lesson plan is really shaping up.
Speaker 2:It's definitely well structured, engaging, and thorough. But, you know, even with the most well designed lessons, there are always a few things that might trip students up.
Speaker 1:Oh, absolutely. It's like those little gotcha moments that make you realize there's a deeper level of understanding needed. So what are some common misconceptions teachers should be prepared for as their students grapple with these falling objects and quadratic functions?
Speaker 2:Well, one that the guide specifically addresses is the idea that the distance fallen is always positive even though the object is moving downwards.
Speaker 1:Which, when you really think about it, makes total sense. It's about the total distance covered, not whether it's up or down.
Speaker 2:You got it. The guide suggest really emphasizing that we're dealing with distance as a magnitude. You know, just how far the object has traveled, not its direction.
Speaker 1:It's amazing how the right language can unlock that understanding. Right?
Speaker 2:Absolutely. It's all about making those subtle but crucial distinctions. And this is where weaving together those different representations, the tables, the graphs, the equations become so powerful.
Speaker 1:It's like having multiple witnesses at a crime scene, each offering a different piece of the puzzle.
Speaker 2:What a fantastic analogy. And by connecting those pieces, those different representations, students can start to build a more holistic understanding of the obtuse motion.
Speaker 1:And hopefully avoid those common pitfalls along the way. So let's recap. We've got notice and wonder to spark curiosity, falling from the sky to uncover that hidden quadratic relationship, and Galileo and gravity to add some historical context and complexity.
Speaker 2:All while anticipating those gotcha moments that might trip students up. It's a well oiled free fall analyzing machine.
Speaker 1:It really is. But where does all of this leave our students in the end? What's the lasting impact of this quadratic deep dive?
Speaker 2:That's where the lesson synthesis comes in. This isn't just about checking off boxes on a lesson plan. It's about solidifying those key takeaways and making sure the learning sticks.
Speaker 1:So we're moving beyond just memorizing formulas and into the realm of true understanding.
Speaker 2:Exactly. The guide really emphasizes the importance of having students articulate their thinking, like, why does the graph look the way it does? How does it connect back to that falling rock?
Speaker 1:Because at the end of the day, math is about making sense of the world around us.
Speaker 2:Couldn't have said it better myself. And to really cement those connections, the guide includes some really insightful discussion questions.
Speaker 1:Oh, like what? Give me an example.
Speaker 2:Well, one question prompts students to compare and contrast those 2 different functions we talked about earlier. The one for distance fallen and the one for distance from the ground.
Speaker 1:So they're really digging into those different perspectives and how they show up in the equations and graphs.
Speaker 2:Right. It challenges them to think critically about why those equations might look different even though they're describing the same event.
Speaker 1:And through that struggle, a deeper understanding of what those equations really represent is born.
Speaker 2:Precisely. But the guide doesn't stop there. It also encourages teachers to connect this lesson back to previous work students might have done with visual patterns.
Speaker 1:Now that's an interesting link. How do those two topics, visual patterns and falling objects, relate to each other?
Speaker 2:At their core, they both involve recognizing and describing patterns, which, as we've seen, can often be represented using mathematical functions.
Speaker 1:Well, it's all connected. Mind blown.
Speaker 2:It's true. This lesson really highlights how math isn't just a bunch of isolated islands, but a beautiful interconnected archipelago of ideas.
Speaker 1:A beautiful interconnected archipelago of ideas. I love that. But there must be some key differences between those visual patterns and our free falling friend. Right?
Speaker 2:You're sharp. You're right. There are. And the guide points out one particularly important distinction. With visual patterns, you typically only deal with whole numbers.
Speaker 2:Right? You can't have half a square in a growing pattern.
Speaker 1:True enough.
Speaker 2:But with a falling object, time is continuous. You can have half a second, a quarter of a second, even a millisecond if you wanna get really precise.
Speaker 1:So we're transitioning from those nice, neat data points of a visual pattern to the smooth, continuous flow of time.
Speaker 2:And that difference is reflected in the graphs. With visual patterns, you're usually working with a series of plotted points. But with our falling object, we get a beautiful continuous curve representing all those infinitesimal moments in time.
Speaker 1:Wow. It's incredible how the type of data we're working with shapes not only our understanding of the math, but also the way we visually represent it.
Speaker 2:Exactly. And by explicitly making that connection, this guide pushes students to think more deeply about functions and their graphs. They're not just static images on a page, they're dynamic representations of real world phenomena.
Speaker 1:We've covered so much ground in this deep dive, sparked some curiosity, built some deep understanding, even made connections to previous learning. But something tells me there's still some uncharted territory to explore. There must be more gold in this lesson plan. Alright. We're back.
Speaker 1:Ready to unearth those final gems from this incredible free fall lesson plan.
Speaker 2:Let's do it.
Speaker 1:It's amazing how much thought goes into creating a lesson that's both exciting for students and gets them to really understand the math.
Speaker 2:It's a real art, isn't it? Mhmm. And this guy does a great job giving teachers the tools they need. But, you know, no matter how detailed the lesson plan is, there are always gonna be those unpredictable moments in the classroom.
Speaker 1:Oh, for sure. Like, when a student asks a question that sends the discussion down a totally unexpected path, teachers need to be ready to
Speaker 2:roll with it. Absolutely. You can't plan for everything, and that's where having a deep understanding of the math yourself is so important. It's not about sticking to a script. It's about feeling comfortable enough with the concepts to explore those spontaneous what if questions.
Speaker 1:Totally. It's like you have your road map, but you're also open to taking a detour if it leads to something interesting. Speaking of interesting paths, this lesson feels like just the beginning of something much bigger. It's like it opens the door to a whole universe of exploration in algebra and physics. What are some other real world phenomena that teachers could tackle using these same mathematical tools?
Speaker 2:Oh, so many possibilities. Just imagine. You could explore the trajectory of a projectile, like a basketball being shot or a rocket taking off.
Speaker 1:Oh, I like where you're going with this.
Speaker 2:Or what about the swing of a pendulum? Even something like population growth over time. Each of these can be modeled and understood using the same principles we've been talking about.
Speaker 1:Wow. So it's like this one lesson gives students a key to unlock patterns across all these different areas. That's incredible.
Speaker 2:Exactly. It highlights how interconnected everything is, and that's what makes this lesson plan so much more than just a lesson on quadratic functions. It's a gateway to a whole new way of seeing and understanding the world through a mathematical lens.
Speaker 1:It's like we've gone from free fall to free thinking. So to all you amazing teachers out there, we hope this deep dive has given you some new ideas and maybe even reignited that passion for bringing this incredible lesson to life in your classrooms.
Speaker 2:Don't forget what the guide says. The ultimate goal isn't just to cover the material, but to uncover that beauty and power of mathematics.
Speaker 1:Could not have said it better myself. A huge thank you to the authors of Illustrative Math for creating such a rich and insightful resource for teachers. And to our listeners, until next time, keep exploring, keep wondering, and keep diving deep.