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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.)
Ready to bring some real world math into the classroom? Always. Today, we're taking a deep dive into a lesson on interpreting and creating graphs. It's lesson 8, interpreting and creating graphs from illustrative math's algebra 1 curriculum.
Speaker 2:Sounds good to me.
Speaker 1:I think this one's really gonna resonate with teachers because it tackles that age old challenge of showing students why graphs matter beyond just plotting points.
Speaker 2:It's true. And this lesson does a fantastic job of achieving that. It's not just about the mechanics of graphing. It's about empowering students to become storytellers through those graphs.
Speaker 1:I love that analogy.
Speaker 2:Okay.
Speaker 1:So how does this lesson guide students to become master storytellers through graphs?
Speaker 2:Well, the lesson plan starts by laying out 3 major learning goals. The first one tackles the concept of average rate of change, and it emphasizes understanding what that rate means in different situations, not just how to calculate it.
Speaker 1:Right. Because we've all had those students who can find the slope of a line but have no clue what it actually represents in the real world.
Speaker 2:Exactly. And that's where this lesson shines. It then shifts to interpreting graph features but not in isolation. Students learn to analyze those intercepts, maximums, and other features within a specific context. They might be looking at a graph and figuring out, okay, what does this point tell us about the height of a flag as it's being raised?
Speaker 1:You've gotta love when math mirrors real life. Okay. So we've got rate of change and interpreting features. What's the 3rd big idea?
Speaker 2:Third one is where it gets really interesting. It guides students to sketch graphs, but instead of giving them equations, it has them work from verbal descriptions or even videos. Imagine the possibilities. You could have them graph the speed of a basketball during a game, the acceleration of a car, or even the temperature of a pot of water as it comes to a boil.
Speaker 1:That's incredible. Not your average plot these points kinda lesson. It sounds like a breath of fresh air for teaching graphing.
Speaker 2:Exactly. It brings the concept to life and reinforces that graphs aren't just these abstract lines on paper. They're powerful tools for representing and understanding the world around us.
Speaker 1:I'm sold. Now I'm curious. How does this all actually play out in the lesson? Walk us through those activities.
Speaker 2:Well, they've got this really clever warm up activity called which one doesn't belong that immediately throws students into the deep end of interpreting graphs.
Speaker 1:Oh, I love a good which one doesn't belong activity. How does this one work?
Speaker 2:They're presented with 4 different graphs, all showing temperature over time, but here's the catch. There are no numbers on the axis.
Speaker 1:Oh, I like that. So they can't just rely on reading specific coordinates. They've to think critically about the overall shapes of the graphs.
Speaker 2:You got it. And the best part is there's no single right answer. Students have to analyze those shapes, look for patterns, and then justify their reasoning. It always leads to a fantastic classroom discussion.
Speaker 1:I can already hear the moments. It's like a detective game, but with graphs. So once they've warmed up their graph analyzing skills, what's next?
Speaker 2:The core of the lesson revolves around a 2 part activity centered around raising a flag.
Speaker 1:Okay. I'm intrigued. Tell me more about this flag raising activity.
Speaker 2:In part 1, students are presented with several graphs, again, without any units labeled. Their challenge is to figure out which of these graphs could realistically represent the height of a flag being raised over time.
Speaker 1:That's brilliant. It forces them to connect the graph to a physical action. They have to think about what's actually possible in the real world. The flag can't just magically teleport upwards. Right?
Speaker 2:Exactly. And to make it even more interesting, part 1 introduces the concept of a vertical line graph. It asks students to really wrestle with the idea of whether a vertical line can actually represent a function.
Speaker 1:Oh, that's a good one. I remember that being a real head scratcher when I first learned it. So how does the lesson approach that concept?
Speaker 2:It doesn't shy away from the challenge at all. It encourages students to debate, question, and really solidify their understanding of what makes a function a function. It's a great opportunity to have them articulate their thinking and justify their answers.
Speaker 1:I love it. Now this is where I think the lesson takes a really fun turn. Right?
Speaker 2:Absolutely. In part 2, students get to put on their graph making hats. They watch a video of a flag being raised, and then their task is to sketch a graph to represent the flag's height over time.
Speaker 1:Wait. So instead of being given data points, they're creating a graph based on what they observe in a video. That's so much more engaging than plotting points from a table.
Speaker 2:Right. And they have to make decisions. Should they represent the height continuously or discreetly? What's a reasonable starting height? How should they scale their axis to fit the data?
Speaker 2:They're applying their knowledge to create a mathematical model of a real world situation.
Speaker 1:It's like they're creating a mini movie, but with graphs and data points instead of cameras and actors.
Speaker 2:Exactly. And let's not forget, they even get to calculate the average rate of change from their own graphs.
Speaker 1:Which brings us right back to that first learning goal. They're not just plugging numbers into a formula anymore. They're seeing how that average rate of change plays out visually and dynamically.
Speaker 2:And that's where the real learning magic happens, when those connections start clicking into place.
Speaker 1:Absolutely. Now I'm sure some of our listeners are thinking, okay, this all sounds great. But what about those inevitable student misconceptions? What are some common roadblocks teachers should be prepared for with this lesson?
Speaker 2:Oh, there are always a few. One that often trips students up is assuming that constant rates automatically mean equal rates.
Speaker 1:Ah, I see what you mean.
Speaker 2:They might look at 2 lines on a graph, both increasing at a steady pace and think, oh, those must represent the same rate of change even if the scenario suggests otherwise.
Speaker 1:Right. Like in that 2 pools activity we talked about earlier with the hoses filling the pools at different rates.
Speaker 2:Exactly. That's a prime example. So how do we help our students avoid that pitfall? Well, emphasizing unit analysis can be a lifesaver here.
Speaker 1:Good point. Make them really think about what those units are telling them.
Speaker 2:Right. Encourage them to break down what each rate actually represents in the context of the problem. Are we talking about the rate of water flowing through the hose? Or are we talking about the rate at which the pool is filling up? Helping them make those distinctions is key.
Speaker 1:It all comes back to connecting those numbers to the real world scenario. Now what other challenges might pop up?
Speaker 2:Another common one, particularly with that bouncing ball activity, is confusing time versus distance on the graph.
Speaker 1:Yeah. I can see how that one could be tricky, especially since the ball is going up and down. That graph might feel a bit more abstract than some of the others.
Speaker 2:Precisely. Emphasizing those axis labels and constantly reinforcing what each axis represents is gonna be crucial. Don't be afraid to be a bit repetitive. Have them explain out loud. Okay.
Speaker 2:The vertical axis shows, and the horizontal axis shows. That explicit connection back to the context will really help solidify the concept.
Speaker 1:It's like giving them a mental checklist to run through as they're interpreting the graph. Now I'm also imagining some students might freeze up when they're asked to sketch a graph without being given specific numbers. You know, starting with a blank slate can feel intimidating.
Speaker 2:Absolutely. It's a natural reaction. But this is where we need to reassure our students that in this lesson, estimation is their friend.
Speaker 1:Yes. We don't need perfect precision here.
Speaker 2:Exactly. The goal isn't to create a perfectly to scale graph. It's about developing their ability to create a reasonable representation of the situation.
Speaker 1:That's a great point. It takes the pressure off and lets them
Speaker 2:focus on the big picture understanding. Now before we move on to
Speaker 1:those optional activities, any other understanding. Now, before we move on to those optional activities, any other expert tips you'd like to share with our listeners? Well, I'd say that this lesson is just brimming with opportunities for rich
Speaker 2:discussion and exploration. So as tempting as it might be to jump in and correct everything, try to resist that urge. Instead, focus on creating a safe and encouraging space for students to share their solutions even if they made different assumptions or took different approaches.
Speaker 1:I love that. It's all about celebrating those light bulb moments when a student figures something out in a way you never even considered.
Speaker 2:Precisely. And whenever possible, find ways to bring those graphs to life. Get creative. Have students physically act out the flag raising or toss a ball around to mimic the bouncing ball. Those visual and kinesthetic connections can make all the difference in how deeply students internalize these concepts.
Speaker 1:It's true. Engagement is key.
Speaker 2:And speaking of connections, this lesson is ripe for tying back to the real world. We want students to see that graphs aren't just stuck in their textbooks. They're everywhere.
Speaker 1:I agree. Showing those real world connections makes it all click.
Speaker 2:To find ways to connect what they're learning to things like stock market trends, weather patterns, even their own growth charts. Help them see those graphs all around them.
Speaker 1:Suddenly, those abstract lines become so much more meaningful and relatable. Okay. So we've covered the core activities of the lesson, but we know teachers are always looking for ways to extend the learning. What can you tell us about those optional activities included in the lesson plan?
Speaker 2:Well, there are 2 additional activities that look particularly intriguing. The first, 2 pools, which we touched on earlier, provides a fantastic opportunity to dive even deeper into the concept of rates.
Speaker 1:Right. With those hoses filling up the pools, what makes this activity stand out?
Speaker 2:What I find particularly effective about this activity is that it doesn't spoon feed students with all the information.
Speaker 1:Oh, so
Speaker 2:it's a bit more open ended. I like it. Exactly. It lays out the scenario hoses filling pools at different rates, but it doesn't specify the exact size of the pools or the flow rates of the hoses. Interesting.
Speaker 2:So students have to fill in some of the blanks themselves? They do. And that's where it gets really good. This open endedness requires students to make assumptions, justify their choices, and really flex those mathematical modeling muscles.
Speaker 1:So there's no single right answer. That's a great way to get them thinking critically and defending their reasoning.
Speaker 2:Precisely. It really pushes their modeling skills. They're not just plotting points. They're creating a mathematical representation of a real world scenario. And speaking of real world scenarios, that second optional activity, the bouncing ball, is a fantastic way to connect graphs to something students can easily visualize.
Speaker 1:Oh, yes. The bouncing ball. It's such a simple concept, but ripe with mathematical potential. How does the lesson use it to explore graphs?
Speaker 2:Well, it goes beyond just graphing the ball's height over time. Students get to experience the data in multiple ways. They see a video of the ball bouncing. They're given still images of the ball at different points in its journey, and they even get a table of values.
Speaker 1:Talk about catering to different learning styles. That's fantastic.
Speaker 2:Right.
Speaker 1:And I imagine giving them that variety of representations really helps solidify the connection between the physical action and the abstract graph.
Speaker 2:Absolutely. And what I find particularly clever is how the activity highlights those key graph features we talked about earlier. Students can actually see the maximum and minimum heights the ball reaches, making those concepts much more concrete.
Speaker 1:It's like those moments when a student realizes that the highest point on the graph corresponds to the exact moment when the ball reaches its peak in the air.
Speaker 2:Exactly. It transforms those abstract points on a graph into meaningful representations of a physical event. And to top it off, the activity even includes a challenge question that prompts students to think critically about the limitations of using still images versus a full motion video.
Speaker 1:Oh, I like that. It gets them thinking about different ways to represent data and which might be more effective depending on what they're trying to show or analyze. That's a really important concept for them to grasp.
Speaker 2:Right. It's subtle, but it's powerful. It reinforces that a graph is just one way to tell a story, and sometimes other methods might be more appropriate or revealing.
Speaker 1:This whole lesson is full of moments for teachers too. I'm already thinking about how I can use these activities in my own classroom.
Speaker 2:Me too. It's so well structured, engaging, and packed with opportunities for students to connect with those big ideas in a meaningful way.
Speaker 1:I couldn't agree more. A huge thank you to the creators of Illustrative Math for developing such a thoughtful and engaging lesson plan. It's definitely going on my list of must try activities.
Speaker 2:Absolutely. It's a fantastic example of how we can bring math to life and make it relevant to our students' experiences.
Speaker 1:So as we wrap up our deep dive into this lesson on interpreting and creating graphs, it really makes you wonder, what other everyday activities could we transform into graph worthy adventures for our students?
Speaker 2:That's the beauty of math. The possibilities are truly endless. It's all about nurturing that curiosity and helping students see the math that's hidden in plain sight all around them.
Speaker 1:I love that. Well said. To all our listeners out there, keep your graphing eyes peeled. You never know what amazing real world connections you and
Speaker 2:your
Speaker 1:students might uncover.