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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.)
Know, one of the toughest things about teaching quadratics is getting students past that initial fear factor.
Speaker 2:Oh, yeah.
Speaker 1:They see x squared. Right. And their brains practically shut down.
Speaker 2:Totally. Like, they've wandered into some higher level math they're not ready for.
Speaker 1:Exactly. So you can imagine how excited I was to dive into this lesson plan, on solving quadratic equations from illustrative math. They've really nailed something important with the way they set up activity 1. Having students grapple with solving an equation like, XXL plus 2 by 35 equals 0 without using factoring.
Speaker 2:Ah, yes. That's such a smart move. Pedagogically, by forcing that initial struggle, and believe me, it it is a struggle to solve that just by plugging in random numbers. They're actually priming the pump for a deeper understanding of why factoring is so valuable.
Speaker 1:So it's not just about finding the answer. It's about making them feel the need for a more efficient strategy.
Speaker 2:Precisely. It's like building their mathematical intuition. They might not even realize it at the time, but that experience is going to make factoring feel like a revelation, not just another rule to memorize.
Speaker 1:And that's exactly what we want. That moment where the light bulb goes off. Mhmm. Now this lesson isn't just about understanding the why behind factoring. It also aims to help students master the how.
Speaker 1:The source material lists 2 main learning targets. One being that students will be able to recognize that quadratic equations can have different numbers of solutions.
Speaker 2:Which is such a critical concept because when they start working with quadratics, it can be easy to assume there's always just one right answer. But by looking at the factored form, they can see how many solutions to expect.
Speaker 1:Okay. Let's unpack that a bit for our listeners.
Speaker 2:Yeah.
Speaker 1:What do we mean by looking at the factored form? So
Speaker 2:let's say you have a quadratic equation that's already been factored, something like, by 5x plus 2 equals 0. Just from looking at those factors, we can see that this equation will have 2 solutions.
Speaker 1:Because each factor could potentially equal 0.
Speaker 2:Exactly. That's the beauty of the 0 product property. And this is where that second learning target comes into play, actually using factored form and that property to solve equations.
Speaker 1:Now I know we, as teachers, understand the 0 product property. But how does this lesson approach it in a way that helps prevent common student misconceptions?
Speaker 2:One thing that's really effective is having students physically manipulate equations.
Speaker 1:Okay.
Speaker 2:Breaking them down into those separate factors and setting each one equal to 0.
Speaker 1:So it's not just about seeing the equation as a whole. Right. It's about getting their hands dirty Yeah. And really understanding the individual components.
Speaker 2:Exactly. And it's about emphasizing that equal to 0 part. Sometimes students rush through and forget that crucial step.
Speaker 1:Right.
Speaker 2:They might see by 5 and think, okay. The answer is 5.
Speaker 1:Oh, I've been there. It's like they forget that we're looking for the value of x that makes the entire equation true.
Speaker 2:Exactly. That's why I love that they included a worked out example in activity 2. Seeing each step laid out clearly with a 0 product property applied methodically can be a game changer for some students.
Speaker 1:Absolutely. Especially when you consider that activity 2 throws in some more challenging examples, like 4 by center set, plus 12 x, plus 9 equals 81. That's not your typical x squared plus something x plus something situation.
Speaker 2:Right. And the source actually points this out as a potential sticking point for students, which I thought was really insightful. It shows that they've anticipated where teachers might need to provide extra support.
Speaker 1:Because let's be honest, even with the best lesson plans, those misconceptions are lurking around every corner, ready to trip our students up.
Speaker 2:Oh, absolutely. And speaking of misconceptions, this lesson plan does a fantastic job of highlighting some common ones.
Speaker 1:Like that classic mistake of moving terms across the equal sign. It drives me crazy when I hear a student say, I'm just gonna move this number over here.
Speaker 2:As if by magic. It's so important to address that head on, reminding them that we're performing operations on both sides of the equation to maintain balance, not just shuffling things around.
Speaker 1:Right. It's like that analogy of a scale. You can't just remove weight from one side without doing the same to the other.
Speaker 2:Precisely. And, you know, another common misconception this lesson cleverly addresses is
Speaker 1:confusing factors with solutions. Oh, tell me about it.
Speaker 2:They see a factor like
Speaker 1:grounds by 3, and it immediately shout out 3. The answer is 3. It's like they get so excited about factoring that they forget to apply the 0 property,
Speaker 2:the idea that one of the factors themselves must equal 0.
Speaker 1:So it's not just about finding the factors. It's about taking that extra step to determine what value of x makes each factor equal to 0. And this is where activity 3 comes in. Right? Connecting those solutions to the graphs of quadratic functions.
Speaker 2:Exactly. And this visual representation is so powerful for solidifying understanding.
Speaker 1:Because when students can actually see that a quadratic equation can have 0, 1, or 2 solutions based on how its graph interacts with the x axis, it clicks in a whole new way.
Speaker 2:Yeah. It's like the difference between hearing a description of a place and actually seeing a photo of it suddenly, it becomes real.
Speaker 1:Exactly. And speaking of making things real, this lesson doesn't shy away from those, gotcha moments that can trip students up. Yeah. It even flags a specific area where students might struggle later on.
Speaker 2:And it's something we haven't even touched on yet. What happens when the coefficient of the x 5 psi term isn't 1? The lesson very deliberately doesn't cover equations like 2x55 plus 5 by 3 even 0, leaving that as an open question.
Speaker 1:Which is brilliant. It plants a seed of curiosity. They've spent this whole lesson mastering a specific type of quadratic equation. And now right when they're feeling confident, you nudge them towards the next challenge.
Speaker 2:Yeah. Yeah.
Speaker 1:It's like that moment at the end of a good book when you realize there's a whole sequel waiting for you.
Speaker 2:Exactly. You're left wanting more.
Speaker 1:This deep dive has been such a fantastic reminder that effective teaching isn't just about presenting information clearly. It's about anticipating those roadblocks, understanding how students learn, and creating those light bulb moments that make the learning stick.
Speaker 2:Couldn't have said it better myself. A huge thank you to the authors of illustrative math for giving us so much to think about.
Speaker 1:Absolutely. Until next time. Keep diving deep into those lesson plans and keep those light bulbs shining bright.