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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 are you ready to dive into some high school algebra?
Speaker 2:I'm always ready for a good algebra deep dive.
Speaker 1:Well, good. Because today, we're getting comfortable with something called vertex form.
Speaker 2:Ah, vertex form. Now we're talking.
Speaker 1:Right. Sounds a little intimidating, but it's really all about quadratic equations. Mhmm. We're actually looking at this lesson plan called lesson 15, vertex form from illustrative mathematics.
Speaker 2:Oh, yeah. I've heard good things about their materials.
Speaker 1:Yeah. It's designed for algebra 1 students. But, honestly, I think anyone who wants to, like, brush up on their algebra skills could benefit from it. For
Speaker 2:sure. And it's not just theoretical stuff either. Understand a vertex form can actually, like, give you some pretty incredible superpower.
Speaker 1:Superpowers. Okay. Now you have to elaborate on that.
Speaker 2:I'm talking about being able to predict the path of a basketball in midair or design the most insane roller coaster loops you can imagine, all thanks to vertex form. Alright.
Speaker 1:You definitely had me at roller coaster. But seriously, those crazy loops and drops all come down to this one equation.
Speaker 2:You got it. It's all about understanding how the different parts of the equation, those parameters, control the shape and position of the parabola.
Speaker 1:So it's like having a secret code to decipher the world of parabolas.
Speaker 2:Exactly. And this lesson is designed to help students crack that code.
Speaker 1:Now, this particular lesson starts with an activity called notice and wonder. Have you heard of that?
Speaker 2:Oh, yeah. Notice and wonder. It's great for getting those brains fired up and making connections.
Speaker 1:Exactly. So they're presented with sets of equations, all representing the same quadratic function, but in different forms. There's factored form, standard form, and, of course, vertex form.
Speaker 2:Ah, so it's all about recognizing those familiar mathematical friends in a new outfit.
Speaker 1:You got it. It's like saying, hey, look. It's the same equation you already know, but dressed up a little differently.
Speaker 2:I like that analogy. It helps students see that vertex form isn't this completely foreign thing. It's just another way of expressing something they're already familiar with.
Speaker 1:Exactly. It's about building those bridges between prior knowledge and this new concept. And speaking of bridges, the lesson then moves into a worked example, showing students how to expand a quadratic expression from vertex form back into that familiar standard form.
Speaker 2:Wait. Wait. Go backward. Why not just stick with vertex form once they've learned it?
Speaker 1:Well, think of it like this. For someone just learning vertex form, it can seem kinda like a foreign language. Right?
Speaker 2:Right.
Speaker 1:So by showing them how it connects back to something they already understand standard form, it helps build that trust and understanding. It's like saying, hey, don't worry. This new thing isn't so scary after all. It's just a different way of writing what you already know.
Speaker 2:Okay. That makes sense. It's all about connecting the dots, showing those relationships between different representations same
Speaker 1:idea. Exactly. Now after all that brain work, it's time to roll up our sleeves and get a little bit visual. Because what's algebra with that little graphing action? I was hoping you were gonna say that because this is where the real fun begins.
Speaker 2:You know it. So the lesson uses a digital graphing tool.
Speaker 1:They recommend Desmos, but, really, any graphing calculator will do.
Speaker 2:Sure. Desmos is great, though, especially for visual learners.
Speaker 1:Oh, absolutely. And with this, they can play around with the different parameters in the vertex form equation.
Speaker 2:Right.
Speaker 1:They change the h value, see how it shifts the graph horizontally. Yeah. Tweak the k value, watch it move up or down.
Speaker 2:It's like giving them a digital etch, a sketch, but for parabolas.
Speaker 1:Love that analogy. Yeah. Because it really is about exploration and discovery. They get to make predictions, test them out, and see those patterns emerge right before their eyes.
Speaker 2:And that kind of hands on experience is so much more engaging than just staring at static equations on a page.
Speaker 1:Absolutely. It brings the math to life. Yeah. Okay. So they've had some fun playing around with the graphing calculator.
Speaker 1:They're starting to get a feel for how those parameters work.
Speaker 2:Yeah.
Speaker 1:What's the next challenge?
Speaker 2:Well, now it's time to see if they can apply what they've learned without the safety net of that calculator.
Speaker 1:Oh, you mean like a pop quiz?
Speaker 2:Sort of. The lesson presents them with a quadratic equation in vertex form and asks them to predict where the vertex will be and whether the graph opens up or down just by looking at the equation.
Speaker 1:Oh, that's a good one. So they have to really visualize it in their minds.
Speaker 2:Exactly. It's like the show me what you know moment. Can they take all those pieces, all those connections they've made, and put them into practice?
Speaker 1:The one thing to recognize a pattern when you see it on a graph Yeah. It's another thing entirely to understand that pattern so well. You can predict it without any visual aids.
Speaker 2:You got it. And that's what this activity is all about. It's about moving beyond rote memorization to a place of deeper understanding.
Speaker 1:And I imagine that's where some of those common misconceptions start to pop up too. Right?
Speaker 2:Oh, absolutely. Like with anything new, there are bound to be a few stumbles along the way.
Speaker 1:So where do things tend to get a little slippery for students? What are some of those common misconceptions that teachers should be prepared for?
Speaker 2:One big one is that h value in the vertex form equation.
Speaker 1:I remember struggling with that one back in the day.
Speaker 2:It's so easy to forget that a positive h actually shifts the graph to the left and not to the right. It feels counterintuitive. Right?
Speaker 1:It does. It's like your brain wants it to go the other way.
Speaker 2:Exactly. And then on top of that, you've got the order of operations to think about, especially when you're expanding that vertex form.
Speaker 1:Right. PEMDAS. Please excuse my dear Aunt Sally.
Speaker 2:There you go. Yeah. I love PEMDAS. But, seriously, that exponent in the vertex form, that can really trip students up if they're not careful.
Speaker 1:It's like a little mathematical landmine just waiting to be stepped on.
Speaker 2:That's a good way to put it. So as a teacher, you really have to emphasize those potential pitfalls and give students plenty of opportunities to practice.
Speaker 1:So besides just drilling them on PEMDAS, what are some strategies teachers can use to help students avoid those vertex form traps?
Speaker 2:Well, one thing that really helps is repetition and visual aids. Have students graph the same equation multiple times, but tweak that h value each time.
Speaker 1:So they could see that shift happening right before their eyes.
Speaker 2:Exactly. It's like a game of hot and cold. Okay. Move the graph to the left a little more. A little more.
Speaker 1:Yeah.
Speaker 2:There you go.
Speaker 1:I love that. It makes it so much more interactive and engaging. And speaking of engaging, let's talk about those real world applications. Because at the end of the day, who wants to just memorize formulas in a vacuum?
Speaker 2:Right. Math is so much more interesting when you can see how it connects to the real world.
Speaker 1:Absolutely. So give us an example. How does vertex form come into play in something like, say, sports?
Speaker 2:Oh, tons of ways. Let's take basketball for instance. You can actually use vertex form to model the trajectory of a basketball shot.
Speaker 1:Woah. Seriously. So you could, like, predict where the ball is gonna land?
Speaker 2:Exactly. You can determine the maximum height of the shot, how far it travels horizontally. All sorts of cool stuff.
Speaker 1:That's amazing. So we're not just giving students a bunch of equations. We're giving them the tools to analyze their favorite sports.
Speaker 2:Precisely. And it's not just sports either. Remember those roller coasters we were talking about?
Speaker 1:Oh, yeah. How could I forget? So vertex form is like the secret sauce behind those gravity defying loops.
Speaker 2:You got it. Roller coaster designers use these principles to calculate the perfect shape and speed of those loops, ensuring maximum thrill while keeping everyone safe.
Speaker 1:Okay. That is seriously cool. Now let's rewind for a second back to that notice and wonder activity at the beginning of the lesson. Why is it so important for students to compare those different forms of quadratic equations?
Speaker 2:That's a great question. And it might seem counterintuitive. Right? If vertex form is the star of the show, why even bother with factored form and standard form?
Speaker 1:Right. It's like, why bring up your x on a first date?
Speaker 2:Well, think of it this way. Those other forms, they're like the foundation. They're what students already know and are comfortable with.
Speaker 1:Okay. So it's like learning a new language. Yeah. You can start picking up individual words and phrases, but it all comes together when you start to see the grammar, the underlying structure that connects it all.
Speaker 2:Exactly. By seeing all those forms side by side students start to notice patterns and make connections, They might realize, hey, that a coefficient seems to be doing the same thing in all of these forms or the way the x is being squared that looks familiar.
Speaker 1:So it's about finding those familiar landmarks in a new city.
Speaker 2:Exactly. It helps them orient themselves within this new form and see how it fits within the larger world of quadratic equations.
Speaker 1:It's like piecing together a puzzle.
Speaker 2:And speaking of piecing things together, remember that worked example where students have to expand the vertex form back into standard
Speaker 1:form? Yeah. It seemed a bit like going backward.
Speaker 2:I know it might seem that way at first, but it's actually a really important step. Remember, for someone seeing vertex form for the first time, it can feel really intimidating.
Speaker 1:Right. It's like being thrown in the deep end when you've just learned to float.
Speaker 2:Exactly. So by showing them how that new scary vertex form can be rewritten in a way they're already comfortable with, it reinforces the idea that this isn't some completely different beast. It's just a different way of expressing the same underlying math.
Speaker 1:It's like saying, hey, look. You can use your existing knowledge to help you make sense of this new thing. You're not cheating.
Speaker 2:Precisely. It builds confidence and shows them that they already have some of the tools they need to tackle this.
Speaker 1:Alright. So we've made those connections. We're feeling a little more confident. Now it's time to fire up those graphing calculators again. Right?
Speaker 1:Mhmm. What makes those graphing activities so powerful for understanding vertex form?
Speaker 2:Well, think about your own learning experiences. Don't things tend to click better when you can actually see them in action?
Speaker 1:Oh, absolutely. I'm a very visual learner. Seeing is believing, as they say.
Speaker 2:Me too. And that's what makes these graphing activities so powerful. Students get to manipulate the different parameters in the vertex form equation that a and instantly see how those changes affect the shape and position of the graph.
Speaker 1:It's like they're conducting their own little visual experiments.
Speaker 2:Exactly. And because they're in control making the changes and observing the outcomes, it leads to a much deeper and more intuitive understanding of how those parameters actually work.
Speaker 1:It's like they're learning to paint with parabolas instead of just looking at them.
Speaker 2:Exactly. And here's the thing, because they're actively involved in
Speaker 1:the process. Right? They're making predictions, testing them out, seeing the result.
Speaker 2:That's how those connections really start to testing them out, seeing the result. That's how those connections really start to solidify.
Speaker 1:So it's not just passive absorption. It's about active engagement.
Speaker 2:Exactly. And one thing I always tell teachers when they're doing these graphing activities is to really encourage that sense of curiosity.
Speaker 1:Right.
Speaker 2:Let students make predictions even if they're wrong.
Speaker 1:Oh, so you're not advocating for right answers all the time?
Speaker 2:Not at all. Sometimes the best learning experiences come from those, wait a minute. Why didn't that work the way I thought it would moments.
Speaker 1:It's like those happy little accidents that Bob Ross talks about. Here
Speaker 2:you go. Those mistakes, they're just opportunities for deeper learning. Encourage your students to analyze their mistakes. Why didn't my prediction match the graph? What can I learn from that?
Speaker 2:And as always, whenever possible, try to connect those mathematical concepts back to something concrete, something in the real world.
Speaker 1:Right. Because who doesn't love a good roller coaster analogy?
Speaker 2:Exactly. But it's not just about making math fun and engaging, though. That's definitely important. It's about showing students the power of vertex form in action. Whether they're analyzing the trajectory of a basketball, designing a roller coaster, or even optimizing profits in a business vertex form gives them this incredibly powerful tool for understanding and modeling the world around them.
Speaker 1:It's like they're unlocking a whole new level of understanding.
Speaker 2:Exactly.
Speaker 1:Now speaking of testing that understanding, let's talk about that activity where students have to put away the graphing calculator and just predict the graph based on the equation alone. That feels like the ultimate test of vertex form mastery.
Speaker 2:It is. That activity is where the rubber really meets the road. Can they take everything they've learned, all those connections they've made, and apply it without relying on visual aids?
Speaker 1:So it's one thing to recognize a delicious looking cake when you see one. It's a whole other thing to be able to bake that cake from scratch.
Speaker 2:Perfect analogy. And for teachers, this activity provides such valuable insight into each student's understanding. Are they confident in their predictions? Are they stumbling over those tricky values? Their responses can really help guide further instruction and ensure that everyone is on solid ground moving forward.
Speaker 1:It's all about meeting students where they are and giving them the support they need to succeed. Well, we've covered a lot of ground in this deep dive into vertex form. We've talked about how this lesson introduces the concept, how it encourages exploration and experimentation, and how it ultimately empowers students to analyze real world situations using this powerful mathematical tool.
Speaker 2:And we've seen how those notice and wonder activities, those graphing explorations, and even those potential pitfalls all play a crucial role in deepening students' understanding of this fundamental concept.
Speaker 1:Absolutely. So as we wrap up, what's the key takeaway you want our listeners to walk away with?
Speaker 2:I think the biggest thing is this vertex form isn't just another abstract formula to be memorized and then forgotten. It's a key that unlocks this deeper understanding of quadratic equations, their graphs, and their real world applications. By approaching it with curiosity, by celebrating those moments, and by connecting it back to the world around us, we can help students see the beauty and power of this really essential mathematical tool.
Speaker 1:It's about making math meaningful, engaging, and relevant.
Speaker 2:Couldn't have said it better myself.
Speaker 1:A huge thank you to the authors of Illustrative Math for creating such a thoughtfully designed and engaging lesson plan.
Speaker 2:It's a fantastic resource for educators, for sure.
Speaker 1:And to our listeners, we hope this deep dive into vertex form has left you feeling energized and maybe even a little bit smarter. Until next time, happy exploring.