{
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  "segments": [
    {
      "speaker": "Speaker 1",
      "startTime": "0.0",
      "endTime": "10.26",
      "body": "Hey, everyone. Welcome back for another deep dive. And today, we're tackling something I know can make some people nervous. Oh, yeah. Quadratic equations."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "10.8",
      "endTime": "11.92",
      "body": "But don't worry."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "11.92",
      "endTime": "13.62",
      "body": "Not gonna lie. A little intimidating."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "13.92",
      "endTime": "18.165",
      "body": "We're gonna break it down with a little help from our friends at Illustrative Math."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "18.385",
      "endTime": "19.205",
      "body": "Always helpful."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "19.345001",
      "endTime": "27.37",
      "body": "They actually have a really interesting approach to teaching this in their algebra 1 curriculum, and luckily for us"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "27.61",
      "endTime": "27.85",
      "body": "Yeah."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "28.01",
      "endTime": "33.63",
      "body": "They've got some detailed notes and lesson plans that give us a peek behind the curtain."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "33.77",
      "endTime": "38.745",
      "body": "I like how they don't just jump right into formulas and equations. Right? They actually take the time to build up to it."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "38.745",
      "endTime": "47.385",
      "body": "Yeah. And speaking of building up to it, they actually mention that students should already have some familiarity with quadratic functions, you know, from, like, a previous unit."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "47.385",
      "endTime": "49.065",
      "body": "It's all about that prior knowledge. Right?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "49.065",
      "endTime": "49.385",
      "body": "Yeah."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "49.385",
      "endTime": "54.63",
      "body": "So students have probably already analyzed graphs, looked at those outputs of quadratic function."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "54.63",
      "endTime": "56.07",
      "body": "Like, maybe projectile motion."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "56.07",
      "endTime": "63.69",
      "body": "Exactly. That's the classic example. They've seen how changing the input affects the output. But here's the twist. This lesson flips the script."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "63.989998",
      "endTime": "69.455",
      "body": "Now it's like, okay. We want a specific output. How do we actually figure out the input that gets us there?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "69.755",
      "endTime": "76.315",
      "body": "That's so interesting. It's, like, they're anticipating that roadblock ahead of time and kinda smoothing it out for students before they even hit it."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "76.315",
      "endTime": "76.815",
      "body": "Exactly."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "76.875",
      "endTime": "79.35",
      "body": "So how do they introduce this whole new way of thinking?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "80.07",
      "endTime": "81.75",
      "body": "Well, with potatoes, of course."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "81.75",
      "endTime": "83.77",
      "body": "Okay. Now you've got my attention. Potato?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "84.31",
      "endTime": "94.725",
      "body": "You heard me right. There's this really fun warm up activity they call the potato problem. Picture this. You've got a potato being launched straight up in the air."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "94.805",
      "endTime": "97.865",
      "body": "Okay. I'm already more engaged than I ever was with parabolas."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "97.924995",
      "endTime": "111.020004",
      "body": "Right. And they give you all the specifics too. Launched from 20 feet up, initial velocity of 92 feet per second, the whole 9 yards. And then students get a graph that shows the potato's height over time."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "111.4",
      "endTime": "114.36",
      "body": "Alright. I can see where this is going. So they've got this graph. Now what?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "114.36",
      "endTime": "119.64",
      "body": "Now the real fun begins. Students have to answer these questions about the potato's flight path."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "119.64",
      "endTime": "121.24",
      "body": "Like what? Give me an example."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "121.24",
      "endTime": "126.605",
      "body": "For example, will this potato ever actually reach a height of, let's say, a 120 feet?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "126.605",
      "endTime": "127.005005",
      "body": "Okay."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "127.005005",
      "endTime": "129.66501",
      "body": "Or even something like, when will it hit the ground?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "130.52501",
      "endTime": "137.58",
      "body": "Those are great questions, and I can definitely see how a graph, while helpful, might not give you the complete picture."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "137.90001",
      "endTime": "144.32",
      "body": "That's exactly the point. It gets students thinking about the limitations of just relying on a graph alone."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "145.02",
      "endTime": "146.88",
      "body": "You might be able to get a rough idea."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "146.94",
      "endTime": "148.14",
      "body": "An estimate, for sure."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "148.14",
      "endTime": "149.52",
      "body": "Yeah. But it's not precise."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "149.58",
      "endTime": "156.315",
      "body": "Not at all. And that's what's so clever about this activity. Mhmm. It highlights that need for something more, something more accurate, something more well"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "156.555",
      "endTime": "158.075",
      "body": "Like a quadratic equation."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "158.075",
      "endTime": "158.875",
      "body": "Boom. Go."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "158.875",
      "endTime": "161.11499",
      "body": "So we've launched potatoes into the air with math."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "161.11499",
      "endTime": "162.575",
      "body": "Hopefully, not real potatoes."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "162.715",
      "endTime": "165.33",
      "body": "Definitely not. And speaking of not real potatoes"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "165.57",
      "endTime": "167.03",
      "body": "We've got those picture frames."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "167.25",
      "endTime": "173.35",
      "body": "Yes. The framing problem. I have to say I love how illustrative math makes these problems so relatable."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "173.65001",
      "endTime": "177.75",
      "body": "Right. It's like, who hasn't struggled to frame a picture on a budget?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "178.125",
      "endTime": "183.745",
      "body": "Seriously. And you can really see how these real world examples tie back to those big mathematical practices."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "183.885",
      "endTime": "187.825",
      "body": "Totally. In the lesson plan, they specifically call out m p 2 and m p 4."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "188.125",
      "endTime": "194.20999",
      "body": "Reasoning abstractly and quantitatively and modeling with mathematics for those who don't have those memorized."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "194.43",
      "endTime": "196.51",
      "body": "Which, let's be honest, is most of us."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "196.51",
      "endTime": "205.62999",
      "body": "True. But the point is those practices are woven into these activities, so students are already flexing those muscles before they even get to the equation part."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "205.62999",
      "endTime": "207.17",
      "body": "It's all about building those connections."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "207.23",
      "endTime": "214.555",
      "body": "Absolutely. So we've got our potatoes. We've got our picture frames. Students are starting to feel the need for a more precise solution."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "214.775",
      "endTime": "217.655",
      "body": "Okay. But hold on. We're not dropping the equation just yet."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "217.655",
      "endTime": "218.775",
      "body": "Oh, right. There's a buildup."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "218.775",
      "endTime": "226.21",
      "body": "It's all about the buildup. Yeah. 1st, they have translate that framing problem, you know, with all the dimensions and materials Yes."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "226.21",
      "endTime": "226.93",
      "body": "All those details."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "227.01",
      "endTime": "228.79",
      "body": "Into a proper mathematical equation."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "228.85",
      "endTime": "236.20499",
      "body": "So it's more about the process of representing the problem with math symbols, not actually solving it at this point."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "236.20499",
      "endTime": "239.965",
      "body": "You got it. And what's cool is there are a few different ways students might approach it."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "239.965",
      "endTime": "245.245",
      "body": "The lesson plan actually mentioned that, didn't it? Like, some students might focus on the length and width of the frame."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "245.245",
      "endTime": "247.965",
      "body": "Right. While others might think about it in terms of area."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "247.965",
      "endTime": "250.5",
      "body": "So they're encouraging different ways of seeing the problem. I"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "251.06",
      "endTime": "254.92",
      "body": "that. Anna, here's the thing. They're not just anticipating different approaches."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "255.06",
      "endTime": "258.28",
      "body": "They're thinking about the things that could trick students up too. Right?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "258.34",
      "endTime": "261.64",
      "body": "Exactly. They specifically call out potential misconceptions."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "262.58",
      "endTime": "264.34",
      "body": "Like what? Can you give an example?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "264.34",
      "endTime": "269.42502",
      "body": "For 1, they mentioned how students might accidentally combine numbers and variables incorrectly."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "269.565",
      "endTime": "271.405",
      "body": "Oh, I can definitely see that happening."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "271.405",
      "endTime": "274.285",
      "body": "Or even mixing up area and perimeter. That's another big one."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "274.285",
      "endTime": "278.625",
      "body": "Those are so common. It's great that the lesson plan addresses those head on."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "278.765",
      "endTime": "282.25",
      "body": "And even better, they offer strategies for tackling them."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "282.25",
      "endTime": "285.53",
      "body": "Scaffolding questions, visual aids, all those good teacher tools."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "285.53",
      "endTime": "288.97",
      "body": "So teachers can feel prepared to guide students through those tricky spots."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "288.97",
      "endTime": "292.545",
      "body": "It's almost like they've got a crystal ball and can see into our future classrooms."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "292.785",
      "endTime": "295.42502",
      "body": "Right. And finally, after all that prep work."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "295.42502",
      "endTime": "300.48502",
      "body": "The big reveal, the quadratic equation in all its glory."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "300.545",
      "endTime": "308.75",
      "body": "Drumroll, please. They introduced the general form, axhacksplusbxpluscplusc where at can't be 0."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "308.89",
      "endTime": "313.69",
      "body": "It's like they've set the stage so well that the equation doesn't seem so intimidating anymore."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "313.69",
      "endTime": "316.97",
      "body": "Because now it's not just some random bunch of letters and numbers."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "316.97",
      "endTime": "318.67",
      "body": "It's got meaning. It's got context."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "318.97",
      "endTime": "324.915",
      "body": "And most importantly, they emphasize that the solutions actually represent something real."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "325.45502",
      "endTime": "329.795",
      "body": "So in our framing problem, a solution isn't just some abstract number."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "330.415",
      "endTime": "333.15",
      "body": "It tells you the actual thickness the frame needs to be."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "333.22998",
      "endTime": "336.35",
      "body": "To use up all your framing material without wasting any."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "336.35",
      "endTime": "338.69",
      "body": "It's like that satisfying moment."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "338.91",
      "endTime": "346.59",
      "body": "I love it. This has been so helpful, but before we get completely lost in the world of picture frames and quadratic equations"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "346.83",
      "endTime": "348.05",
      "body": "We gotta wrap it up."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "348.435",
      "endTime": "354.595",
      "body": "Do we actually see how to solve these equations in this particular lesson, or is that for another time?"
    },
    {
      "speaker": "Speaker 2",
      "startTime": "354.595",
      "endTime": "358.135",
      "body": "They leave us hanging just a bit, which honestly is kind of brilliant."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "358.275",
      "endTime": "361.255",
      "body": "A little math cliffhanger to keep us coming back for more."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "361.315",
      "endTime": "361.815",
      "body": "Exactly."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "362.035",
      "endTime": "365.15002",
      "body": "So they leave us in suspense, but they don't leave us high and dry."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "365.21002",
      "endTime": "367.05002",
      "body": "They give us a little something to think about. Right?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "367.05002",
      "endTime": "369.23",
      "body": "A little homework assignment for next time."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "369.61002",
      "endTime": "376.03",
      "body": "Exactly. It's like they wrap up by asking, how could you take this these ideas and make them work for your own students?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "376.345",
      "endTime": "380.745",
      "body": "Okay. So not just how to solve the equations, but how to actually teach it."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "380.745",
      "endTime": "382.605",
      "body": "It's about taking those big ideas"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "382.985",
      "endTime": "383.145",
      "body": "Yeah."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "383.305",
      "endTime": "386.665",
      "body": "You know, connecting to things they already know, making it fun and engaging"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "386.905",
      "endTime": "389.085",
      "body": "Right. Building up that need for the equation."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "389.27",
      "endTime": "390.79",
      "body": "And then letting teachers run with it."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "390.79",
      "endTime": "395.13",
      "body": "It's almost like they're saying, we've given you the tools. Now go build something amazing."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "395.19",
      "endTime": "399.85",
      "body": "Because at the end of the day, who knows their students poorer than the teachers themselves?"
    },
    {
      "speaker": "Speaker 1",
      "startTime": "399.91",
      "endTime": "404.725",
      "body": "That's so true. I feel like we could talk about this all day, but sadly, all good things must come to an end."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "404.725",
      "endTime": "405.705",
      "body": "They really do."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "405.765",
      "endTime": "416.345",
      "body": "And this has been such an interesting deep dive. I feel like I learned a lot, not just about quadratic equations themselves, but about how to approach teaching them in a way that actually sticks."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "416.645",
      "endTime": "423.18002",
      "body": "I think that's what I love about illustrative math. They really put a lot of thought into the why behind the what."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "423.24002",
      "endTime": "430.38",
      "body": "Couldn't have said it better myself, so to Illustrative Math, thank you. And to everyone listening, thank you for joining us on another deep dive."
    },
    {
      "speaker": "Speaker 2",
      "startTime": "430.44",
      "endTime": "431.56",
      "body": "Until next time."
    },
    {
      "speaker": "Speaker 1",
      "startTime": "431.56",
      "endTime": "432.64",
      "body": "We'll see you then."
    }
  ]
}