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.)
Alright. Get ready to dive into linear equations. Not just how to solve them, but how to make them, like, meaningful, you know, for your students. We're really going to take apart lesson 11, connecting equations to graphs, part 2 from illustrative math.
Speaker 2:Oh, this is gonna be good.
Speaker 1:And trust me, this is about way more than just plotting points.
Speaker 2:Yeah. What's really exciting about this lesson is how it helps students connect the dots literally between, like, what's on the page and what they see in the real world. You know?
Speaker 1:You know me. I love a good connection. Yeah. So how does this lesson kick off? Like, give me the warm up routine.
Speaker 2:Okay. So they start by rewriting expressions like 10+42. Okay. And, 642 seems basic. Right?
Speaker 1:Right.
Speaker 2:But it's actually sneaky prep for rearranging linear equations later on. It's all about the distributive property.
Speaker 1:Sneaky. I like it.
Speaker 2:Which, let's be honest, can trip students up even in later grades.
Speaker 1:Especially with those negative signs lurking around. Isn't that, like, a common pitfall the lesson even calls out?
Speaker 2:Definitely. Definitely. It's so easy to, like, divide the 10 by the 2, but totally forget to also divide that 4.
Speaker 1:Yep. Yep.
Speaker 2:Emphasizing that both parts of the expression need to be divided is crucial. That's where, like, that solid understanding starts.
Speaker 1:It's like making sure everyone at the party gets a slice of pizza. Mhmm. No term gets left behind. But, okay, they flex their algebra muscles. Now what?
Speaker 1:How does the lesson bring graphs into the picture but, like, in a compelling way?
Speaker 2:Okay. So this is where things get really clever. They use 2 scenarios from previous lessons.
Speaker 1:Okay.
Speaker 2:1 about a water tank draining
Speaker 1:Uh-huh.
Speaker 2:And the other about buying almonds and figs.
Speaker 1:Interesting.
Speaker 2:The students already have some familiarity with these, but now they're challenged to connect, write specific parts of the equations
Speaker 1:Okay.
Speaker 2:To the graphs Mhmm. And the real world scenarios.
Speaker 1:That's like they're looking for clues, but the clues are hidden in plain sight, like, within the equations themselves. Right. Tell me more about how this plays out.
Speaker 2:For example, in the water tank problem, they might notice that the equation has a never a 20 in it. The lesson prompts them to consider what that never a 20 represents on the graph. It's the rate at which the water is draining.
Speaker 1:So they start to see that those numbers aren't just arbitrary. They have, like, real world meaning. It's like the equation because a translator between the math and, like, the actual situation.
Speaker 2:Exactly. And that's where those light bulb moments happen, where students move beyond just, like, rote memorization, and they start to really grasp the concepts.
Speaker 1:Yes. Love that. But knowing my students, they're itching for some hands on action. What's next in the illustrative math lesson plan?
Speaker 2:Get ready for the slope match challenge.
Speaker 1:Oh, I love a good challenge.
Speaker 2:Imagine a stack of cards, right, with linear equations in their standard form, like, say, 2x +3y206.
Speaker 1:Okay.
Speaker 2:Each student gets an equation and has to find their match another card showing the slope and y intercept of that equation.
Speaker 1:Oh, that sounds like a really fun way to, like, shake things up. It's like a mathematical scavenger hunt. Right. But I'm guessing there's more to it than just matching. Right?
Speaker 1:Yeah. What are some strategies students might use to, like, tackle this challenge?
Speaker 2:That's where it gets really interesting because there are multiple ways to approach this. Right?
Speaker 1:Okay.
Speaker 2:Some students might rearrange the equation to get it into that familiar y
Speaker 1:equals mx plus b form.
Speaker 2:Right. Right. Others, y equals mx plus b form.
Speaker 1:Right. Right.
Speaker 2:Others might be able to just look at the equation as is
Speaker 1:Uh-huh.
Speaker 2:And figure out the slope and intercept just by analyzing the coefficients.
Speaker 1:Oh, wow.
Speaker 2:And some might even use a quick graphing method.
Speaker 1:So many options. Right. I love how this one activity allows for all those different entry points and showcases, like, the strategic thinking involved in math.
Speaker 2:Absolutely. And for teachers, this presents a fantastic opportunity for, like, rich classroom discussion.
Speaker 1:Oh, yeah. For sure.
Speaker 2:Imagine asking students, okay. You found your match. Now tell me how you got there. What strategy did you use? Why?
Speaker 1:Yes. It's not just about getting the right answer. It's about understanding the process and being able to, like, articulate your thinking. Now that's what I call deep learning. But before we dive deeper into those discussions, let's take a quick breather.
Speaker 1:We'll be back in a flash to uncover even more treasures from this illustrative math lesson. Alright. So we're back. And we've seen how this illustrative math lesson, like, cleverly connects equations, graphs, real world scenarios.
Speaker 2:It's really cool.
Speaker 1:But let's get real for a second. Yeah. We know students often hit those, like, inevitable roadblocks. Totally. What are some common misconceptions teachers should, like, watch out for?
Speaker 2:One word, negatives. Oh, yeah. They can be sneaky little gremlins, especially when students are rearranging equations.
Speaker 1:Mhmm.
Speaker 2:The lesson even highlights this potential pitfall where students might divide only part of the equation by a negative instead of all the terms.
Speaker 1:That's like they forget to share the negative sign with everyone at the party.
Speaker 2:Exactly. They're
Speaker 1:hoarding all the negativity.
Speaker 2:Yes. Reinforcing that every term needs its fair share of that negative sign is key. And it all circles back to that, like, trusty distributive property we talked about earlier.
Speaker 1:It's all connected.
Speaker 2:Right.
Speaker 1:I love how this lesson keeps reinforcing those fundamental algebraic principles.
Speaker 2:Absolutely. And speaking of connections, another potential stumbling block is understanding, like, the meaning behind those letters in the slope intercept form. Yeah. You know, y equals mx plus b.
Speaker 1:Ah, yes. The old what does m stand for again? Dilemma.
Speaker 2:Right. This is where, like, clear visuals can be a game changer.
Speaker 1:Oh, for sure.
Speaker 2:The lesson even suggests using, like, color coding to help students visually connect those kinda abstract symbols to their meaning.
Speaker 1:I'm all about color coding. Mhmm. Imagine highlighting for slope in one color. Mhmm. Maybe even drawing a little slanted line on the board to really, like, drive home that concept.
Speaker 2:Yes.
Speaker 1:And then use a different color for b, the y intercept, maybe drawing a big dot where the line crosses the axis.
Speaker 2:I love that. Making those visual connections can really solidify understanding.
Speaker 1:For sure.
Speaker 2:And it's not just about the visuals, but also about constantly bringing it back to the context of the problem. Right.
Speaker 1:What does the slope actually represent in this specific scenario? What does the intercept tell us about the starting point?
Speaker 2:It's about making sure those letters and numbers on the page actually mean something to students. Right? It's about bringing that math to life.
Speaker 1:Yes. 100%.
Speaker 2:And this lesson does a fantastic job of doing just that.
Speaker 1:It really does.
Speaker 2:But it doesn't stop there.
Speaker 1:It goes deeper.
Speaker 2:It goes beyond just linear equations and gets really meta when you say.
Speaker 1:Absolutely. It pushes teachers and students to think about, like, the bigger picture of mathematical understanding. Yes. The lesson ends with this thought provoking question.
Speaker 2:Okay.
Speaker 1:How can we extend these ideas to other types of equations?
Speaker 2:That's what I'm talking about. It's like they're planting these seeds for, like, future mathematical exploration.
Speaker 1:Exactly. It's about encouraging that what if. Thinking Yes. If understanding the structure of a linear equation unlocks so much, what about quadratic equations? What about exponential equations?
Speaker 2:The possibilities are endless.
Speaker 1:And it all starts with building those foundational skills and connections.
Speaker 2:Totally.
Speaker 1:But before we get ahead of ourselves, let's pause here for a moment.
Speaker 2:Okay.
Speaker 1:We've still got more to unpack from this treasure trove of a lesson. Alright. We're back for, like, the final stretch of our deep dive into linear equations.
Speaker 2:The home stretch.
Speaker 1:We've talked about connecting equations to graphs, real world situations, even hinting at those, like, broader applications.
Speaker 2:Right.
Speaker 1:But what are some, like, concrete takeaways teachers can actually use in the classroom with this illustrative math lesson?
Speaker 2:Well, the lesson itself gives some great pointers.
Speaker 1:Okay. I like it. For
Speaker 2:instance, when working with those standard form equations
Speaker 1:Mhmm.
Speaker 2:Like 2x plus 3y is 6 Okay. They suggest this really clever nudge for students. Hey. What happens when x is 0? What about when y is 0?
Speaker 1:Oh, I see where you're going with this. Right. Without even rearranging the whole equation Yeah. You can lead students right to the intercepts.
Speaker 2:Exactly.
Speaker 1:If x is 0, boom, you've got your y intercept
Speaker 2:There
Speaker 1:it is. And vice versa.
Speaker 2:It's like magic, but it's not. Right? It's a little shortcut that comes from, like, understanding that deeper structure of the equation.
Speaker 1:Yes.
Speaker 2:And it can really get students thinking strategically.
Speaker 1:I love those little moments you can create in the classroom.
Speaker 2:Oh, the best.
Speaker 1:It's not about magic. It's about, like, giving them the tools to see the patterns for themselves.
Speaker 2:Empowering them.
Speaker 1:Yes. And the lesson also emphasizes being mindful of the language we use.
Speaker 2:Oh, absolutely.
Speaker 1:Instead of just saying, like, solve for y Right. Which can feel a bit mechanical.
Speaker 2:Just robotic, we might say. Let's rewrite this equation so that y is isolated.
Speaker 1:It's subtle, but it changes the whole feel. We're not just, like, following a set of rules.
Speaker 2:Right.
Speaker 1:We're on a mission to, like, transform this equation and, like, reveal his secrets.
Speaker 2:Unlock its potential.
Speaker 1:Yes.
Speaker 2:It's about purpose, not just procedure.
Speaker 1:I love that.
Speaker 2:And the beauty is these skills, they're building the rearranging, the connecting to graphs, the real world applications. It's all setting them up for success with more complex functions down the road.
Speaker 1:Quadratics, here we come.
Speaker 2:Bring it on.
Speaker 1:It really is all connected.
Speaker 2:It all ties together.
Speaker 1:But before we wrap things up, this illustrative math lesson got my brain buzzing with one final thought. Okay.
Speaker 2:I'm ready.
Speaker 1:We've talked a lot about how understanding the equation helps us understand the graph. Right?
Speaker 2:Right.
Speaker 1:But what if we flip the script?
Speaker 2:Oh, okay. I like where this is going.
Speaker 1:Could understanding the graph of a linear equation actually help us make predictions about the equation itself?
Speaker 2:Interesting.
Speaker 1:Could we work backwards? Like, using the visual clues to, like, piece together the algebraic representation.
Speaker 2:That is such a great point. It adds a whole new layer of, like, depth and intrigue. Right. Suddenly, we're not just plotting points. We're, like, mathematical detectives using every clue at our disposal to crack the case.
Speaker 1:Exactly. And that, my friend, is the beauty of math.
Speaker 2:It really is.
Speaker 1:It's not just about finding the answer. Mhmm. It's about exploring, like, the infinite connections and possibilities that lie within every concept.
Speaker 2:So well said.
Speaker 1:A huge thanks to illustrative math for giving us so much to ponder and to our listeners. Keep those mathematical minds sharp and remember, sometimes the most enlightening discoveries happen when we dare to look at things from a different perspective. Love it. Until next time.