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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 dig into some algebra 1?
Speaker 2:Let's do it.
Speaker 1:Alright. We are tackling factoring quadratics in this deep dive.
Speaker 2:Okay.
Speaker 1:Specifically, lesson 10
Speaker 2:Alright.
Speaker 1:Where things get really interesting. You know that feeling when you're teaching and suddenly you see the light bulb go off for a student?
Speaker 2:Oh, yeah.
Speaker 1:That's what we're going for today. Absolutely. Helping your students have that moment with factoring.
Speaker 2:It's incredibly satisfying. Yes. And this lesson tackles a concept that often trips students up.
Speaker 1:Okay.
Speaker 2:Those quadratics where the coefficient of the squared term isn't 1. Right. We're gonna give you the tools to guide your students through this, and we'll even plant the seed that factoring, while useful, isn't always the most efficient tool in our algebra toolbox, a little foreshadowing for future lessons.
Speaker 1:I love that setting them up for success later.
Speaker 2:Exactly.
Speaker 1:So this lesson plan from Illustrative Math starts with a clever warm up activity.
Speaker 2:Okay.
Speaker 1:Which one doesn't belong? It throws 4 quadratic expressions at your students right out of the gate.
Speaker 2:Right.
Speaker 1:Some in factored form, some in standard form.
Speaker 2:Right.
Speaker 1:I love how this gets them thinking critically about the structure of quadratics before you even introduce the new material. Yeah. It's like a sneak peek into the world they're about to enter. Yes. This activity is also a fantastic opportunity for you to gauge their existing knowledge and identify any misconceptions they might have.
Speaker 1:Yes.
Speaker 2:As you listen to their explanations, pay close attention to their mathematical vocabulary.
Speaker 1:Oh, absolutely.
Speaker 2:Are they using terms like linear term or coefficient Yeah.
Speaker 1:Correctly? Right.
Speaker 2:Are they recognizing the different forms an expression can take? These observations will help you tailor your instruction to their needs.
Speaker 1:It's like a detective game Yes. Listening for clues in their language.
Speaker 2:Exactly.
Speaker 1:Then the lesson dives into the heart of the matter. Okay. The activity has students expand factored expressions like 2x plus 1, x plus 4. And then the tables turn, they need to figure out how to reverse the process. Yeah.
Speaker 1:It's in this reversal that many students will have that moment realizing that factoring becomes more nuanced when that leading coefficient isn't half. This is where having a strong grasp coefficient isn't half. This is
Speaker 2:where having a strong grasp of the distributed property becomes essential. Yes. Encourage your students to explain how those factors in
Speaker 1:factored
Speaker 2:form connect to the coefficients in standard form. Right. And don't be surprised if they resort to some guess and check at first.
Speaker 1:Yeah. It's a natural starting point. But as you can imagine, it can get messy quickly, especially with more complex expressions. Oh, absolutely.
Speaker 2:This is where you can start hinting at the need for more reliable and efficient methods, which, don't worry, we'll explore in future lessons.
Speaker 1:Okay. So we're building up to some more powerful tools.
Speaker 2:We
Speaker 1:are. But first, let's ground this in a real world scenario.
Speaker 2:Okay.
Speaker 1:The lesson plan does a great job of this with the timing of blob of water problem.
Speaker 2:Okay.
Speaker 1:Students get to apply what they're learning to calculate how long it takes for a water droplet from a fountain to hit the ground.
Speaker 2:Okay.
Speaker 1:Sounds fun. Right?
Speaker 2:Great.
Speaker 1:But here's the catch. The quadratic equation they'll come up with doesn't factor easily using rational numbers.
Speaker 2:Right.
Speaker 1:Time for a reality check.
Speaker 2:Exactly. And this failure to factor neatly presents a fantastic learning opportunity.
Speaker 1:Right.
Speaker 2:It challenges the assumption that all quadratic expressions can be factored simply.
Speaker 1:Yes.
Speaker 2:And it sets the stage for introducing more robust techniques like graphing.
Speaker 1:Okay.
Speaker 2:You can demonstrate how graphing provides an approximate solution when factoring hits a roadblock, even if the solution isn't a neat integer.
Speaker 1:Right. So it's all about having the right tool for the job. Yes. And sometimes we need something more powerful than our trusty factoring techniques.
Speaker 2:Exactly.
Speaker 1:Now for those teachers who have students ready for an extra challenge or maybe up for challenging yourself, the illustrative math lesson plan includes an optional activity on the substitution method.
Speaker 2:Right.
Speaker 1:Have you used this one before?
Speaker 2:I have. Yeah. The substitution method is like a bridge between basic factoring and more advanced techniques. It involves temporarily replacing the squared term with a new variable to make that leading coefficient 1.
Speaker 1:Okay.
Speaker 2:This allows for simpler factoring before substituting back the original variable.
Speaker 1:So it's like putting on a disguise to make the problem easier to work work
Speaker 2:with. Exactly. I like it. But keep in mind, this method requires a solid grasp of algebraic manipulation and an understanding of why each step is taken. Yes.
Speaker 2:It's not just about memorizing a procedure, but about truly comprehending the underlying concepts.
Speaker 1:Right. We want that deep understanding.
Speaker 2:Yes.
Speaker 1:This feels like a good place to pause and reflect.
Speaker 2:Okay.
Speaker 1:We've covered a lot of ground in this first part of our deep dive.
Speaker 2:Yeah.
Speaker 1:What are some key takeaways you'd like our listeners to keep in mind as we move
Speaker 2:forward? Well, I think the main takeaway so far is that while factoring is a powerful tool for solving quadratic equations, it's not a one size fits all solution.
Speaker 1:Right.
Speaker 2:Recognizing this limitation is crucial for students as it opens their minds to exploring other methods, which is something this lesson does very well.
Speaker 1:Yeah. It's like that saying, if all you have is hammer, everything looks like a nail.
Speaker 2:Exactly. Now as you guide your students through this lesson, be prepared for some potential wait what moments.
Speaker 1:Okay.
Speaker 2:A common misconception is that if the first and last terms match up when expanding their factored form, then they've got it right.
Speaker 1:Okay.
Speaker 2:They might completely forget about that sneaky middle term.
Speaker 1:Oh, yeah. It's like putting together a puzzle and only focusing on the corners.
Speaker 2:That's a great analogy. To counter this, encourage them to carefully distribute those factors and check their work against the original expression, making sure all the pieces fit perfectly.
Speaker 1:Yeah. And speaking of puzzles, what about when that leading coefficient isn't a perfect square?
Speaker 2:Right.
Speaker 1:The illustrative math lesson actually includes an optional activity to address this. Okay. They introduced this idea of multiplying the whole
Speaker 2:activity to address this. Okay. They introduced
Speaker 1:this idea of multiplying the whole expression
Speaker 2:by a strategic
Speaker 1:form of 1 to temporarily transform that leading coefficient into a perfect square, which can then be factored more easily. Right. What have you seen when you've used that in the classroom?
Speaker 2:This strategy often throws students for a loop because it involves a higher level of abstract reasoning.
Speaker 1:Right.
Speaker 2:They might wonder, why are we allowed to multiply by something out of nowhere?
Speaker 1:Yeah.
Speaker 2:This is where you step in to emphasize that multiplying by 1 in this clever disguise doesn't change the expression's value. It just dresses it up in a way that makes it easier to work with.
Speaker 1:K. So it's like a mathematical magic trick.
Speaker 2:Precisely with a purpose.
Speaker 1:However, it's important to be mindful that not all students will be ready for this level of abstraction. Right. You know, your student's best. So use your judgment to determine if this optional activity will enhance their understanding or lead to unnecessary confusion.
Speaker 2:Right. Use it strategically. Yes. Great point. Now let's circle back to the timing a blob of water problem.
Speaker 1:K.
Speaker 2:Remember, this real world scenario highlights a crucial concept.
Speaker 1:Yeah.
Speaker 2:Not every quadratic expression will factor neatly using integers.
Speaker 1:Right.
Speaker 2:This is a big moment for many students, so it's important to address it head on.
Speaker 1:What advice do you have for teachers when their students hit this wall of unfactorability?
Speaker 2:Transparency is key.
Speaker 1:Okay.
Speaker 2:Let your students know that not all quadratic expressions will factor easily, and that's perfectly okay.
Speaker 1:Right.
Speaker 2:Remind them that real world scenarios are often messy and don't always fit neatly into integer solutions. Yes. This is where having other tools in our problem solving toolbox, like graphing and eventually the quadratic formula becomes invaluable.
Speaker 1:Right. It's about empowering them to embrace the challenge and explore alternative approaches.
Speaker 2:Absolutely. You can even turn these factoring failures into opportunities for discovery.
Speaker 1:Oh, I love that.
Speaker 2:Encourage students to use technology to graph these trickier quadratics and observe where the graph intersects the x axis.
Speaker 1:Okay.
Speaker 2:This visual representation provides a sneak peek of the solutions even if those solutions are irrational numbers.
Speaker 1:So we're connecting the abstract world of algebra to the visual world of graphs. I love it.
Speaker 2:It's all connected. Yeah. And this visual representation can often spark deeper understanding and curiosity.
Speaker 1:Right. It gives them that visual Yeah.
Speaker 2:Now before we wrap up this deep dive into factoring quadratics, what's one key message you'd like to leave our listeners with as they prepare to teach this lesson? It's like those times in life when a wrong turn leads you to an even more amazing destination.
Speaker 1:Exactly. Those what just happened moments can actually be incredibly powerful learning experiences. Now before we wrap up, we wanna leave you with a little something extra to ponder as you dive into this lesson plan. The illustrative math team does a fantastic job of scaffolding the learning account, but what if you could take it a step further? How could you extend this lesson to incorporate real world data that might lead to quadratic equations that can't be factored easily?
Speaker 2:Oh, I love that challenge. What kind of real world scenarios could we use?
Speaker 1:Well, projectile motion's a classic example. You could have students analyze data from a tennis ball toss or the trajectory of a basketball shot. The quadratic equations that model these situations often involve decimals and won't factor neatly.
Speaker 2:That's a great idea. It would really drive home the point that the real world doesn't always fit into neat little boxes. Right. And it shows students that while factoring is a useful tool, it's not the only tool. By introducing them to these messier real world scenarios, you're preparing them for the complexities they'll encounter in more advanced math and science courses.
Speaker 1:It's about building that bridge between the textbook and the real world.
Speaker 2:Exactly. And who knows? Maybe you'll even inspire a future physicist or engineer in the process.
Speaker 1:That would be amazing. Yeah. Well, on that note of inspiration, we wanna give a huge shout out to the brilliant minds at Illustrative Math for creating such a well crafted and thought provoking lesson plan.
Speaker 2:Absolutely.
Speaker 1:This has been a fantastic deep dive
Speaker 2:It is.
Speaker 1:Into factoring quadratics.
Speaker 2:Yeah.
Speaker 1:I always learn so much from you. I really appreciate your insights.
Speaker 2:Well, thank you for having me.
Speaker 1:And to all our listeners out there, thank you for joining us for this deep dive into algebra 1.
Speaker 2:Yes. Thank you.
Speaker 1:We'll see you next time for another deep dive into the world of math education.