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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.)
Ever feel like your students are just memorizing steps when it comes to functions? Like, they're missing the y behind the graphs and the equations?
Speaker 2:Yeah.
Speaker 1:Today's deep dive is for you.
Speaker 2:Okay.
Speaker 1:We're tackling illustrative math's lesson 12 on piecewise functions.
Speaker 2:Alright.
Speaker 1:And we're going beyond just summarizing the lesson. Okay?
Speaker 2:Okay.
Speaker 1:We're uncovering the why behind their teaching choices so you can confidently guide your students to a deeper understanding.
Speaker 2:That's right. Because true mastery in math comes from understanding the concepts, not just memorizing formulas.
Speaker 1:Right.
Speaker 2:And that's where this lesson really shines.
Speaker 1:Okay. So let's unpack this. Okay. Imagine you're explaining to a student how to calculate the cost of parking based on how long they've parked.
Speaker 2:Right.
Speaker 1:The first hour might cost one price, the next few hours, another
Speaker 2:Sure.
Speaker 1:And then a different rate after that.
Speaker 2:Yeah. You've just hit on a perfect example of a piecewise function in action without even realizing it.
Speaker 1:Really?
Speaker 2:It's a rule book with different instructions for different input ranges.
Speaker 1:Now I love that illustrative math doesn't just throw the abstract concept at students. Right. They've got that frozen yogurt scenario. Yeah. What's the strategy behind easing in like that?
Speaker 2:It's all about building on prior knowledge and making connections relatable experience like buying frozen yogurt by weight, students can intuitively grasp how the price changes based on how much yogurt they get.
Speaker 1:Mhmm.
Speaker 2:This groundwork sets the stage for understanding those pieces we talked about
Speaker 1:Uh-huh.
Speaker 2:And how they relate to mathematical rules.
Speaker 1:So it's almost like they're internalizing the concept before even seeing the formal notation.
Speaker 2:Exactly. When you introduce the abstract later, it's not this foreign thing.
Speaker 1:Yeah.
Speaker 2:It's simply a way to represent what they already understand on an intuitive level. Yeah. This approach is backed by research. You know? Studies show that when students can connect mathematical concepts to real world situations, their understanding and retention improve significantly.
Speaker 1:That makes sense. It's like learning a new language. You pick up the everyday phrases before you dive into grammar rules.
Speaker 2:Right.
Speaker 1:Speaking of which, let's talk about a bit of grammar. In the world of piecewise functions, those open and closed circles on the graphs Yeah. It feels like those can really trip students up.
Speaker 2:They can be tricky. It's true.
Speaker 1:Yeah.
Speaker 2:The postage stamps activity in the lesson is a good way to approach this.
Speaker 1:Okay.
Speaker 2:Think about it. You can't have a letter that weighs 0 ounces. Right? Right. That real world constraint helps illustrate why we use an open circle when a specific input value is excluded.
Speaker 1:So the open circle is, like, visually representing that strictly less than situation.
Speaker 2:Precisely.
Speaker 1:Okay.
Speaker 2:It's a visual key that highlights the boundary of the function.
Speaker 1:And a closed circle means we're dealing with less than or equal to. That input value is fair game.
Speaker 2:You got it.
Speaker 1:Okay.
Speaker 2:That closed circle tells us that particular input value has a defined output within that piece of the function. The lesson does a good job of emphasizing the link between the visual representation on the graph and the inequality symbols used in the function's definition.
Speaker 1:So they're really driving home that connection between the graph and the algebraic representation.
Speaker 2:Absolutely. And this is where that deeper understanding can really prevent those common misconceptions from taking root.
Speaker 1:You're right. Because let's face it, even with the best explanations
Speaker 2:Right.
Speaker 1:Students can still hit those roadblocks.
Speaker 2:Yeah.
Speaker 1:So let's jump into those misconceptions, the source material highlights.
Speaker 2:Okay.
Speaker 1:The first one that stood out to me was confusing which circle to use when graphing. Yeah. Have you encountered that in your own teaching experience?
Speaker 2:Definitely. It's easy for students to get those open and closed circles mixed up
Speaker 1:Right.
Speaker 2:Especially when they're first grappling with the concept.
Speaker 1:It's like they understand the idea, but then those circles become their nemesis on the graph. Right. What's your go to strategy for clearing that up?
Speaker 2:I find that consistently bringing it back to the inequality symbols and their meaning helps when students are deciding on an open or closed circle.
Speaker 1:Yeah.
Speaker 2:I have them underline or highlight the or equal to part of the inequality if it exists.
Speaker 1:Okay.
Speaker 2:That visual reminder often clicks for them. Yeah. Sometimes literally drawing a door open or closed on the point on a number line
Speaker 1:Okay.
Speaker 2:Can connect the concrete to the symbolic. Love that.
Speaker 1:Yeah. Those visual cues can be so powerful.
Speaker 2:Right. Another misconception
Speaker 1:the source brings up is students defining intervals with overlapping inputs.
Speaker 2:Yeah.
Speaker 1:I could see where they'd stumble there.
Speaker 2:It's a natural mistake to make, especially as they're getting comfortable with the notation. Right. They might accidentally create rules where multiple parts of the function could apply to the same input value.
Speaker 1:Which mathematically is a big no no.
Speaker 2:Exactly. That's where those real world connections become even more valuable.
Speaker 1:Okay.
Speaker 2:I like to go back to our parking lot example. I'll ask, could you imagine if the parking lot charged you 2 different prices for the same hour you were parked?
Speaker 1:That would be chaos.
Speaker 2:Exactly. And it helps them see the logical inconsistency in their thinking. Right. When they can see that their mathematical model doesn't make sense in the real world, it prompts them to reexamine their work and understand why overlapping inputs aren't allowed.
Speaker 1:It's all about making those connections. Right? Showing them that math isn't just some abstract set of rules. Yeah. It's a tool for understanding the world around them.
Speaker 2:Exactly.
Speaker 1:Now once students have a good handle on the basics, illustrative math introduces the bike sharing activity
Speaker 2:Yeah.
Speaker 1:Which feels like a real step up in complexity.
Speaker 2:It is.
Speaker 1:What makes this activity so effective in your opinion?
Speaker 2:This activity shifts the learning from interpretation to application.
Speaker 1:Okay.
Speaker 2:Instead of simply interpreting provided graphs and equations, students have to generate their own table, graph, and d, a verbal description based on a given set of rules for a bike sharing scenario.
Speaker 1:So they're really taking ownership of the concept at this point?
Speaker 2:Precisely. They're moving beyond simply recognizing piecewise functions.
Speaker 1:Right.
Speaker 2:They're creating and representing them in multiple ways. Okay.
Speaker 1:This activity also encourages communication and collaboration, which are essential skills in math and beyond.
Speaker 2:It's like they're becoming fluent in the language of piecewise functions.
Speaker 1:Yes.
Speaker 2:The source mentions, though, that this activity can be a bit more challenging because it introduces non constant linear expressions.
Speaker 1:Right.
Speaker 2:Why do you think that trips some students up?
Speaker 1:Up until this point, most of the examples they've encountered have involved horizontal lines on the graphs Okay. Representing constant rates.
Speaker 2:Okay.
Speaker 1:The bike sharing activity introduces the idea that pieces of a piecewise function can be lines with slopes other than 0.
Speaker 2:So visually, the graphs become a bit more complex.
Speaker 1:Exactly. And for some students, visualizing and plotting non constant linear expressions can be a bit more challenging.
Speaker 2:So for students who are used to just plotting points, seeing those slanted lines might feel like uncharted territory.
Speaker 1:That's a great way to put it. It's like they've been navigating with a compass, and suddenly, they're handed a map with contour lines.
Speaker 2:K. I love that analogy.
Speaker 1:So how do we guide them through that transition from points to those more complex lines?
Speaker 2:This is where visual aids become especially crucial.
Speaker 1:Okay.
Speaker 2:Graph paper, rulers, different colored pencils
Speaker 1:Mhmm.
Speaker 2:Those can be game changers.
Speaker 1:Mhmm.
Speaker 2:Even manipulatives like building blocks can help students grasp the idea of a changing slope.
Speaker 1:Yeah. Yeah.
Speaker 2:It's about providing those concrete tools to support their developing understanding of the abstract concepts.
Speaker 1:That's why we're giving them the scaffolding to build those mathematical connections.
Speaker 2:Right.
Speaker 1:Now before we wrap up, I wanted to circle back to something you mentioned earlier about how this lesson isn't just about understanding what a piecewise function is, but also how it connects to broader mathematical ideas.
Speaker 2:Right.
Speaker 1:The source specifically calls out domain and range. Yeah. Could you elaborate on that a bit more?
Speaker 2:Absolutely. Domain and range are fundamental concepts that extend far beyond just piecewise functions.
Speaker 1:Okay.
Speaker 2:Thinking back to our bike sharing example, the domain would be all the possible rental times. Right?
Speaker 1:So the input, how long someone might rent a bike for?
Speaker 2:Exactly. And the range would be all the possible rental costs Okay. The output
Speaker 1:Yeah.
Speaker 2:Based on those times.
Speaker 1:I see where you're going with this. Yeah. We're taking those abstract ideas of domain and range and grounding them in the context of a real world scenario that students can wrap their heads around.
Speaker 2:Precisely. And that's what makes this lesson so powerful.
Speaker 1:Right.
Speaker 2:It's not just about teaching students how to graph piecewise functions.
Speaker 1:Right.
Speaker 2:It's about equipping them with the tools and understanding to apply those concepts to real world situations, to model relationships, and to see the interconnectedness of mathematical ideas.
Speaker 1:It's like we're giving them a lens through which they can view and analyze the world around them.
Speaker 2:Exactly. And hopefully, sparking that curiosity to keep exploring and making those connections.
Speaker 1:That's what it's all about. Yeah. Well, on that note, I think we've successfully navigated the intricacies of Piecewise Functions today.
Speaker 2:We did it.
Speaker 1:A huge thank you to the authors of Illustrative Math for providing such a well crafted and thought provoking lesson and to you, our listeners, for joining us on this deep dive.
Speaker 2:It's been a pleasure.
Speaker 1:Until next time. Keep those mathematical conversations flowing.