Subscribe
Share
Share
Embed
Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Ready to dive into some algebra 1?
Speaker 2:Always up for an equation or 2.
Speaker 1:Today, we're looking at algebra 1 lesson 5, how many solutions from illustrative mathematics. I know from experience that quadratic equations can really throw students for a loop, especially when it comes to figuring out how many solutions there are.
Speaker 2:Yeah. It's not always obvious, is it?
Speaker 1:Not at all. So before we get into the nitty gritty of the lesson plan, let's take a step back. Why is understanding the number of solutions so important?
Speaker 2:Well, think of it this way. It's about more than just algebra. It's about problem solving.
Speaker 1:Interesting. Tell me more about that.
Speaker 2:Okay. So let's say a student's using a quadratic equation to model something in the real world, like, I don't know, the trajectory of a basketball.
Speaker 1:Okay. I like where you're going with this.
Speaker 2:Understanding the number of solutions helps them figure out if their answer makes sense. Like, are there multiple possibilities, just one solution, or maybe even no solutions at all?
Speaker 1:Right. Because sometimes what they're trying to do is just impossible, and the equation shows that. It's like the equation is giving them a reality check.
Speaker 2:Exactly. This lesson isn't just about the math itself. It's about building those critical thinking skills.
Speaker 1:I see. So it's about connecting those graphs Yeah. Equations and real world scenarios.
Speaker 2:Absolutely. And this lesson plan does a great job of that by walking students through 3 main ways to find the number of solutions.
Speaker 1:Oh, you mean, like, the 0 product property?
Speaker 2:Exactly. And, of course, there's graphing.
Speaker 1:Of course.
Speaker 2:But then there's my favorite reasoning.
Speaker 1:Reasoning, you mean figuring it out without even needing a graph?
Speaker 2:Yep. It's like giving them X-ray vision for algebra.
Speaker 1:I love that. X-ray vision for algebra. I might have to steal that one. But speaking of x-ray vision, should we use ours to take a peek inside this lesson plan?
Speaker 2:Let's do it.
Speaker 1:So let's jump into this warm up activity, math talk, 4 equations. Looks like it's all about those true false statements.
Speaker 2:It's a clever way to get those gears turning early on. Gets them thinking about squaring negatives, the 0 product property, all that good stuff.
Speaker 1:It's like a little sneak peek to see what they already know.
Speaker 2:Right. Like, that one statement asks if negative 5 squared equals 25 means that negative 5 is a solution.
Speaker 1:Oh, yeah. I see it.
Speaker 2:It seems simple, but you'd be surprised how many students stumble on that one.
Speaker 1:They forget about the bigger picture of the entire equation.
Speaker 2:Exactly. It's a good reminder that details matter. Speaking of details, let's talk about solving by graphing.
Speaker 1:Good old graphing. A classic for a reason. Right?
Speaker 2:Absolutely. There's something so powerful about seeing that visual representation.
Speaker 1:It just clicks for some students.
Speaker 2:Seeing those x intercepts where the graph crosses the x axis.
Speaker 1:That's where y equals 0.
Speaker 2:Right. It suddenly makes those x values, the solutions, seem so much more real.
Speaker 1:It's not just an abstract idea anymore. Okay. So next up is finding all the solutions. Now this is where it gets fun. Right?
Speaker 2:Right. Because now students get to choose their own approach, graphing or reasoning.
Speaker 1:Graphing or reasoning, like picking your weapon of choice.
Speaker 2:I like that. And the choice itself is important. It helps them develop that metacognition.
Speaker 1:Thinking about their own thinking. So meta.
Speaker 2:Exactly. And don't forget that little, are you ready for more section, always love a good challenge.
Speaker 1:Gotta keep those advanced students on their toes. Okay. Now this next activity, analyzing errors in equation solving. This one really resonates with me.
Speaker 2:Oh, yeah.
Speaker 1:Pria and Diego's mistakes. So typical. Trying to use the 0 product property when the equation doesn't equal 0, dividing both sides by a variable. It's like looking in a mirror at my own students.
Speaker 2:Classic mistakes, but such valuable teaching moments. Speaking of which, maybe we should delve a bit deeper into those common misconceptions, you know, really unpack them.
Speaker 1:Yes. Let's do that. I think understanding those misconceptions is key to helping students succeed in algebra. I think the 0 product property is a perfect example. Students learn it, and then boom, they wanna use it for everything even when it doesn't apply.
Speaker 2:It's like they forget that whole thing about multiplication and 0.
Speaker 1:Right. If you're multiplying a bunch of numbers and get 0, one of them has to be 0.
Speaker 2:Exactly. So when they see an equation like x minus 12 times x minus 6 equals 7, and they try to set each factor equal to 7
Speaker 1:It's a dead giveaway that they're missing that connection.
Speaker 2:Totally. And then there's the other classic dividing both sides of the equation by a variable.
Speaker 1:Tell me about it. Seems like a shortcut, but it can really mess things up.
Speaker 2:Big time. Remember Diego?
Speaker 1:Oh, yeah. Poor Diego.
Speaker 2:He thinks he's simplifying, but he's actually making a solution vanish into thin air.
Speaker 1:It's like a magic trick, but not the fun kind. We need those solutions to reappear.
Speaker 2:And that's where graphing can be so helpful. It shows them what's happening visually.
Speaker 1:Because when they divide by a variable, they're basically, like, erasing part of the graph.
Speaker 2:The perfect way to put it. And who knows what they erased? Maybe a crucial intersection point.
Speaker 1:Exactly. Speaking of crucial, let's not forget about that squaring misconception.
Speaker 2:Oh, yeah. The old, I only see one solution when there are really 2.
Speaker 1:They get so focused on the positive side of things, they forget about its negative twin.
Speaker 2:It's like the number line only has one side for them. Yeah. So an equation like x equals 9, they'll say x equals 3, and that's it.
Speaker 1:It's like they've never even met negative 3. Okay. So we've talked about the common traps. Now how do we equip teachers to help their students avoid them? What can they do in the classroom to make this stick?
Speaker 2:Well, for the 0 product property, I say make it a mantra before they even think about factoring.
Speaker 1:Get them with the
Speaker 2:Set it equal to 0. Set it equal to 0.
Speaker 1:I love it. Make it a rule, a game. Repetition is key. Right?
Speaker 2:Absolutely. And for the variable division dilemma.
Speaker 1:What about that one?
Speaker 2:Make checking their work a nonnegotiable.
Speaker 1:Oh, such a good habit.
Speaker 2:It's not just about finding an answer. It's about making sure it actually works. Plug those solutions back in. If it doesn't work, back up and try again.
Speaker 1:Exactly. And then, hopefully, a light bulb goes off. Wait. Maybe I shouldn't have divided by that variable. Okay.
Speaker 1:Last one. Those squared equations and their missing negative solutions. Any tips for those?
Speaker 2:Visuals. Visuals. Visuals. Graphing is your best friend here.
Speaker 1:Show them. Don't just tell them.
Speaker 2:Right. When they graph an equation like x dot equals 9, they'll see those two beautiful points where the graph hits the x axis, then it clicks.
Speaker 1:Two intersection points. Two solutions. It all makes sense.
Speaker 2:And it's not just about memorizing a rule. It's about understanding why it works.
Speaker 1:It all comes back to those connections. Right? This lesson plan really does a fantastic job of guiding teachers and students through those.
Speaker 2:It really does. Provides a solid foundation for, you know, even more advanced methods later on, like the quadratic formula.
Speaker 1:Oh, the quadratic formula. We could do a whole another deep dive on that one.
Speaker 2:For another day.
Speaker 1:Exactly. But for today, I think the biggest takeaway for our listeners is this, anticipate those misconceptions. Emphasize the 0 property. Make them check their work and use visuals. Give your students those tools, and they'll be quadratic equation masters in no time.
Speaker 2:Couldn't have said it better myself. Remember, these concepts might seem small, but they build on each other. It's all connected to the bigger picture of math.
Speaker 1:So true. Well, on that note, a huge thank you to Illustrative Mathematics for this awesome lesson plan. And to our listeners, as you gear up to teach algebra 1, lesson 5, how many solutions. Oh. Remember, you've got this.
Speaker 1:With a little preparation and a whole lot of enthusiasm, you can guide your students to conquer those quadratic equations.
Speaker 2:Algebra. More like Al g s.
Speaker 1:Okay. I think we've reached our quota for algebra puns for one day.
Speaker 2:Agreed.
Speaker 1:Until next time.