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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 caught yourself wondering how on earth anyone could use math to map out the crazy cool arc of a frog jumping or maybe even a pumpkin being launched through the air? Yeah. Me too. Wait. This that's exactly what we're diving into today, folks.
Speaker 1:Quadratics. We're taking a look at graphs that represent situations straight out of the illustrative math playbook. And let me tell you these presenter notes. They're packed with gold nuggets on how to make these equations click for our students.
Speaker 2:You know what's really neat about this lesson? It takes those quadratic equations, the ones that often leave students wondering, when will I ever use this, and flips the script making them feel so relatable.
Speaker 1:Totally. It's like instead of just memorizing formulas, they're cracking the code, uncovering the real world stories hidden within those numbers and graphs.
Speaker 2:Exactly. It's all about connecting the dots, showing them how the same mathematical relationship can be expressed in a bunch of different ways, the equation, the graph, and to get the the actual real life situation itself. That's where the magic happens,
Speaker 1:right, when it clicks and they see those connections. So let's talk about those parabolic curves we see all over this lesson. They're everywhere. What can we, as teachers, really emphasize to help students grasp what each part of that parabola actually represents?
Speaker 2:I think it's key to remind them that those parabolas, they're not just pretty shapes. You know, they're telling a story. Take the x intercepts for instance. Those points show us exactly where the graph intersects with that horizontal axis, which often translates to something really concrete, like when an object hits the ground.
Speaker 1:So, like, if we're talking about a frog leaping into the air, which by the way is totally an example in this lesson.
Speaker 2:Then those x intercepts, they show us the precise moments when that frog leave the ground and when it lands again.
Speaker 1:Okay. I'm with you. And what about that peak, the highest point on the curve? What's the vertex telling us?
Speaker 2:The vertex, that's a gold mine of information. It gives us the lowdown on the maximum or minimum value. Sticking with our froggy friend, that vertex, that's the highest point of its jump.
Speaker 1:Wow. So just by looking at the graph, you instantly know how high that frog jumped and how long the whole shebang lasted just from those two key points.
Speaker 2:Exactly. And that's just scratching the surface, and we're only looking at the graph. Now imagine diving into the equations themselves, both in standard and factored forms. They add a whole other layer of depth to that understanding.
Speaker 1:Okay. Because if I'm being honest, sometimes those equations can look a little intimidating even for us teachers. Right?
Speaker 2:Oh, absolutely. But the key is to remember, they're just another way, more mathematical way to describe that same parabolic curve we're seeing on the graph. So
Speaker 1:it's like if the graph tells the story visually, the equation tells it. How would you even describe that?
Speaker 2:Mathematically. And those different forms of the quadratic equation, like the standard form, you know, x plus plus x plus c equals 0, It helps students grasp how those different forms connect to the visual of the graph and what each piece of that equation truly represents in a real world scenario.
Speaker 1:So we've got these different forms of quadratic equations. Right? Like, different ways of writing them down. What can we actually do with them? What do they tell us about the parabola itself, and how does that connect to all those cool real world situations the lesson's focusing on?
Speaker 2:Totally. Let's break it down starting with the big kahuna, the standard form, you know, x plus bx+0. 0.
Speaker 1:Oh, yeah. That one.
Speaker 2:It might look like a jumbled mess at first glance, but trust me, those coefficients, a, b, and c, they're like secret codes just waiting to be cracked.
Speaker 1:I like the way you think. Spill the beans. What secrets are we talking about? Alright.
Speaker 2:So that a
Speaker 1:value, it's kind of the boss. It tells us whether our parabola is
Speaker 2:gonna be a happy camper, opening upwards like a big old smile, or if it's gonna be a bit more down in the dumps, opening downwards like a frown.
Speaker 1:Which visually can represent so many things. Right? Like a ball flying through the air or the curve of a bridge.
Speaker 2:Exactly. And then we've got our trusty c's value. That's our y intercept, the point where our parabola decides to cross paths with the I axis.
Speaker 1:And in those real world scenarios, that often represents a starting point.
Speaker 2:Bingo. Like, picture this. You're launching a rocket. Right. And that launch pad, it's perched 10 feet up in the air.
Speaker 2:Well, guess what? The c in your equation, that's gonna be 10.
Speaker 1:Okay. That makes sense. Yep. Now we've talked about the standard form. What about the factored form?
Speaker 1:Why is it helpful for students to understand this form as well as the standard form?
Speaker 2:Think of the factored form as putting on a new pair of glasses, you know. It gives us a fresh perspective on that equation. It's like this. It lays bare the equation's roots, those crucial x intercepts we talked about earlier.
Speaker 1:So instead of zeroing in on that single highest or lowest point, the vertex Mhmm. We're highlighting where the graph makes a b line for the x axis.
Speaker 2:Right. And those points, they often line up with those key moments of starting and stopping in the real world, like, when that rocket takes off and when it decides to call it a day and heads back down.
Speaker 1:And seeing those key moments right there in the equation, that's gotta be powerful for students. It's like the equation is giving you a sneak peek into the action.
Speaker 2:Exactly. It takes all the guesswork out of it. You see the equation and bam, you know exactly where those critical points are.
Speaker 1:This is making so much sense. We've got the graphs giving us that visual and then the equations in their different get ups providing those precise mathematical details. But how do we actually make these abstract ideas come alive for our students? Because let's be real, sometimes math can feel a bit, well, abstract.
Speaker 2:That's where those engaging activities come in. And trust me, they're designed to make those moments practically unavoidable.
Speaker 1:Okay. Color me intrigued. Spill the tea. What's one of those activities that really get students fired up about quadratics?
Speaker 2:Alright. So one that always comes to mind is this one called a jumping frog. It's brilliant in its simplicity. Students get their hands on a quadratic equation, but here's the kicker. It's modeling the height of a frog's jump over time.
Speaker 2:Their mission, should they choose to accept it, is to dissect that equation, figure out when that frog decides to leave the ground, when it makes its grand reentrance, and just how high that froggy daredevil manages to soar.
Speaker 1:Love it. It's not just about crunching numbers anymore. It's like you're a fraud scientist or something.
Speaker 2:Exactly. They're solving for x, but they're also figuring out what x actually means in the context of a frog living its best life midair.
Speaker 1:It gives it purpose. Okay. That's awesome. Speaking of awesome scenarios, what about that flight of 2 baseballs activity? What's the deal with that one?
Speaker 2:Okay. So imagine this. You've got 2 baseballs taking flight. Classic. Right?
Speaker 2:But here's the twist. Students get hit with different representations for each baseball's journey. One's all laid out in a neat little graph, while the other, they're gonna have to decode that bad boy from an equation.
Speaker 1:Oh, sneaky. So they have to put their quadratic function knowledge to the test comparing these 2 aerial adventures even though the information is coming at them in totally different forms.
Speaker 2:You got it. It's like they're on a top secret mission to crack the quadratic code. And the best part, it pushes them to think outside the box about quadratic functions and really hammers home that crucial link between the visual world of the graph and the algebraic world of the equation.
Speaker 1:It's like they're learning to speak different mathematical languages fluently. I'm loving it. Yeah. Now you also mentioned another activity that sounds super interactive, something called info gap, rocket math. That one sounds intriguing.
Speaker 2:That one's a winner. It's all about teamwork and communication. 2 things that are absolute gold both in and out of the classroom. So picture this, we've got our students paired up ready to tackle some rocket science. But here's the catch, Each student only gets half the information they need to solve the problem.
Speaker 2:It's like a mathematical mystery.
Speaker 1:Talk about a recipe for disaster or maybe a recipe for some serious collaboration. They gotta rely on each other, communicate like pros to piece together the missing parts of the puzzle.
Speaker 2:You nailed it. 1 student might be sitting there with an equation that describes a toy rocket's grand launch while their partners got their hands on the rocket's maximum height.
Speaker 1:But to crack the case, to solve the problem, they've gotta talk to each
Speaker 2:other, ask the right questions, crack the case, to solve the problem, they've got to talk to each other, ask the right questions, listen carefully, and really wrap their brains around the information they're getting and giving.
Speaker 1:Okay. It's officially a mathematical puzzle where communication is key. It's not just about getting the right answer. It's about the journey, the back and forth, and explaining your thought process along the way.
Speaker 2:And you know what? That's where the real magic happens. Those moments when students make those connections through collaboration.
Speaker 1:I can already see how these activities can make learning about quadratic functions so much more engaging and meaningful for students. But I'm also wondering about the flip side. What about the challenges? I mean, even with the best design lessons, math can sometimes throw us a curveball. Right?
Speaker 2:Oh, absolutely. There are always a few common misconceptions that tend to trip students up, especially when they're first dipping their toes into the world of quadratic functions.
Speaker 1:Yeah.
Speaker 2:And you know what? That's exactly what we're gonna explore next.
Speaker 1:Yeah. So easy to get swept up in those light bulb moments. Yeah. You know? But in the real world of teaching math especially, there are always a few of those concepts that tend to leave students scratching their heads.
Speaker 2:Oh, tell me about it. Even with the most engaging activities, you can dream up those pesky misconceptions. Misconceptions. They always seem to find a way to sneak in.
Speaker 1:Okay. So let's shine a light on those potential stumbling blocks, those things that make them go, wait a minute. What's one area where you see students hit a wall with quadratic functions?
Speaker 2:You know one that pops up all the time is how to make sense of negative results, especially when we're talking about real life situations.
Speaker 1:I see what you mean. It's like you're looking at a graph of a basketball's arc after a free throw, and the equation throws a negative value for time at you.
Speaker 2:Exactly. And students, they're left wondering, hold on. How can time be negative? Did we accidentally invent a time machine?
Speaker 1:Right. It's like trying to rewind real life.
Speaker 2:And that's where that link back to the real world is so crucial. We can explain that, yeah, mathematically, the equation might go into the negative time zone. But in this particular problem, we're only interested in what happens after the ball leaves the player's hand.
Speaker 1:So it's more about equipping them to understand the boundaries of the model.
Speaker 2:Precisely. The math gives us this big picture, but we've gotta zoom in on the part that lines up with our specific problem.
Speaker 1:That's a really good point. Now what's another one of those gotchas that teachers might run into in the classroom?
Speaker 2:Another tricky one is connecting those different ways we can represent a a quadratic function. Students might totally get it when they see a parabola on a graph, but then they hit a wall trying to see how that same idea can be shown with an equation, whether it's in standard form or factored form.
Speaker 1:It's like they're looking at the same idea, but it's spoken in different languages. Mhmm. And they haven't quite become fluent in how to translate between them.
Speaker 2:I love that analogy. And to help bridge that gap, visuals are key. Encourage teachers to have their students graph the same quadratic function in both standard and factored form.
Speaker 1:So they can literally see that even though the equations look different, they end up creating the same parabolic curve on the graph.
Speaker 2:Exactly. It's like this visual confirmation that, It's the same relationship just wearing different outfits.
Speaker 1:I like it. Different outfits for different occasions.
Speaker 2:Right. And to really drive that point home using those real world examples can be super powerful.
Speaker 1:Give me an example. How would you do that?
Speaker 2:Let's go back to that factored form. When students see it, they can immediately pick out those x intercepts. If they're thinking about a ball being tossed in the air, those points show exactly when the ball makes contact with the ground.
Speaker 1:It's all about forging those concrete connections. Yeah. Helping them to see that those equations as abstract as they might seem are really just snapshots representations of real things actually happening.
Speaker 2:And that's the beauty of this lesson. It takes that abstract world of math and brings it to life. It shows students that, hey, this stuff, this is real.
Speaker 1:We've covered a ton of ground in this deep dive. From those oh so elegant equations to those hands on activities, from potential pitfalls to strategies for sparking those moments.
Speaker 2:And we've seen firsthand how a well crafted lesson can equip teachers to guide their students towards a deeper, more meaningful understanding of quadratic functions.
Speaker 1:And a huge shout out to the brilliant minds at Illustrative Math for giving us such a valuable resource. As we wrap things up here, we wanna leave you with something to ponder. We've explored how quadratic functions can map out the path of a projectile, but that's just the tip of the iceberg. What other real world mysteries could you and your students unlock using the tools from this lesson? Until next time, keep those deep dive explorations going strong.