Series: Collaborating to Develop Mathematical Ideas


Common core State Standards

  • Math:  Math
  • HSF-IF:  High School: Functions: Interpreting Functions
  • B:  Interpret functions that arise in applications in terms of the context
  • 5: 
    Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Download Common Core State Standards (PDF 1.2 MB)

Modeling & Graphing Real-World Situations
Lesson Objective: Create distance-time graphs
Grades 9-12 / Math / Functions

Thought starters

  1. How can mathematical modeling increase engagement?
  2. What structures does Ms. McAtee have in place for supporting all students?
  3. How does Ms. McAtee use formative assessment?
I really like the way Ms. McAtee allows students to go through multiple phases of thinking about graphing the ball-toss situation, encouraging them to discuss their ideas, pose their own questions, and modify their thinking and their graphs. By allowing students to work in pairs and small groups, students can discuss the task at hand and build upon one another's ideas. They come to realize that it's okay not to know the "answer" in the beginning - it takes time to fully understand the mathematics behind real world situations. All students and all questions are given respect by the teacher - she acknowledges students' thinking without making negative judgments - she supports their ways of thinking when they are able to provide acceptable justifications, thereby increasing their confidence in their own abilities to think mathematically. This, in turn, encourages them to engage more deeply with the content and build on their own understandings. Great job!
Recommended (1)
I really like this for our Social Skills class time. The multiple phases of thinking, the working in pairs and discussing their own ideas. Showing students that it's okay not to know the "answer", but that working together and discussing it will help them understand and prepare them for what's behind the real world situations.
Recommended (0)
Here's a good resource
Recommended (2)
I really have been trying to increase my proficiency in teaching the concept of functions to my students. I often have felt frustrated by students’ lack of attention to detail in the components of the graph. This lesson and the way it is designed give students a reason to want to pay attention to the detail. The videos capture their attention. The collaboration keeps them engaged and provides them with contrasting ideas when there is a detail they might miss. You have helped me to see how I can scaffold and expand this work for my students by having them construct graphs of these relationships themselves. Being responsible for creating these graphs and comparing work with their peers, highlights details students might otherwise miss. Thank you!
Recommended (0)
I have been looking for tasks where students can find success while persevering in problem solving. I appreciate how Ms. McAlee encourages students to attempt different approaches to interpret the problem. Ms. McAlee did not dismiss the student who started the graph with a value of zero. Instead she encouraged the student to think of how that argument would work. This showed how to implement Standard for Mathematical Practice #3, "Construct viable arguments and critique the reasoning of others." While the focus may be F-IF.5, I find that this video really emphasizes how to incorporate the Standards for Mathematical Practice continually. I would like to know of other tasks for different common core standards that offer students the opportunity to engage in theses Math Practices. Thank you for sharing this task!
Recommended (0)


  • Modeling & Graphing Real-World Situations Transcript
    GFX: Tch Teaching Channel

    +++ 00:00:07 +++
    Krista McAtee: I’m going to give

    Modeling & Graphing Real-World Situations Transcript
    GFX: Tch Teaching Channel

    +++ 00:00:07 +++
    Krista McAtee: I’m going to give you a second to go ahead and graph. And remember, this is a sketch.
    Student: It doesn’t go higher each time.
    Student: It goes like this.
    Student: It goes up, take three seconds to go to another graph and three seconds--
    Krista McAtee: The most important thing is to have a task or an activity where everybody has access to.
    Illustrative Mathematics: Modeling Real World Functions
    Krista McAtee: If you remember on Thursday, we did the water line activities, right? And we have lots of cool graphs that you all made.
    Lower third:
    Krista McAtee, High School Math Teacher, Sonoma Valley High School, Sonoma, CA

    +++ 00:00:39 +++
    Krista McAtee: My name is Krista McAtee. I teach at Sonoma Valley High School in Sonoma, California. And I teach a course called Bridge to Geometry, which is a course for kids who have passed algebra or resource algebra and are not quite ready to enter mainstream geometry.
    Krista McAtee: Take a look at that, how it’s filling up there. My question for you is this. Why is it that the graph looks like that? The graph is this green line right here.

    +++ 00:01:10 +++
    Student: Because the cup isn’t very tall.
    Krista McAtee: Okay. So the water line doesn’t go up very high. So our heighth line can’t go very high.
    Krista McAtee: They’re a group of kids who generally feel like they are awful at math and don’t want to risk or try anything in math. So that’s kind of how they come to me in the beginning of the year.
    Krista McAtee: I’m going to show you one other person’s and on this piece of paper that you have there, I would like for you to draw what you would guess. And this is just a sketch.

    +++ 00:01:48 +++
    Krista McAtee: We’re focusing on functions and having students understand look at a mathematical situation and being able to graph that situation.
    Krista McAtee: Okay. Take a couple minutes and discuss what you’ve graphed with your partner and see if you agree on more or less the sketch of your graph.

    +++ 00:02:08 +++
    Student: The chart’s at a steady pace going up rather slowly and then it shoots up and then gradually goes higher and then the top widens out and goes up slowly.
    Student: Yeah. As it gets to the smaller part, it moves up a lot quicker.
    Student: Yes.
    Krista McAtee: This is a mathematical modeling task. So understanding that water being poured into jugs and balls being thrown into the air, it has something to do with math. And making that connection between what’s happening in the world and being able to create a model of that thing that’s happening in the world.

    +++ 00:02:46 +++
    Krista McAtee: Okay. So I want you to look carefully at this video.

    Krista McAtee: You’re going to check out what these teachers are doing. And then I’m going to ask some questions.
    Krista McAtee: The next part of our lesson, we looked at a video of two of my colleagues that I’ve been collaborating with where they playing catch with a softball.
    Krista McAtee: What are some things that we could graph that you notice? We’ve been doing graphing of the water. What could we graph?

    +++ 00:03:13 +++
    Leo: I saw when the ball when it’s going back and forth kind of like a slope.
    Krista McAtee: Okay. So you’re talking about how we could graph it. What would we graph?
    Leo: I don’t know.
    Krista McAtee: Heighth still, right? We would be talking about the heighth of what? The heighth. Is that what you were talking about, Leo, when you were kind of visualizing what that would look like?
    Leo: Yeah.
    Krista McAtee: Okay.

    +++ 00:03:34 +++
    Student: The graph would make the shape of a parabola.
    Krista McAtee: Okay. So you’re thinking about which graph would? The heighth over time?
    Student: Yes, the heighth.
    Krista McAtee: Okay. I’m going to play this again and you’re going to draw how you think the heighth of the ball is over the time of the video. And the video is a minute.
    Krista McAtee: Then we decided to graph heighth. So I had them individually graph the heighth of the ball over time.
    Krista McAtee: One graph of the whole time, the heighth over the whole time.

    +++ 00:04:14 +++
    Krista McAtee: So this is the-- what’s on this--
    Student: That’s time.
    Krista McAtee: This is time. So at the beginning, they’re throwing it high, right?
    Student: Yeah.
    Krista McAtee: And then later on is when they’re throwing it lower.
    Student: Yeah.
    Student: Oh. Yeah, yeah, so I get that. That’s like--
    Student: So it’s high and then it just drops and then just straight.
    Krista McAtee: Okay.
    Student: Yeah. You know what I mean?
    Krista McAtee: You could-- yeah, try it.
    Student: Okay.

    +++ 00:04:35 +++
    Krista McAtee: Formative assessment plays a role in my classroom all the time. I am constantly wandering around and monitoring what kind of work that they’re doing so I can go oh, this is the question I need to ask next or if they’re all having a-- or several of them are having a misunderstanding about one thing, then I’ll pull that up and I’ll ask on the overhead or whatever. And I’ll say, “What do you guys think about this? Does everybody agree with this?” And it allows for students to revise their thinking.

    +++ 00:05:03 +++
    Krista McAtee: Can you come and show what you did? And you guys see what parts of what Leo has to say that you agree with and maybe if you have any questions for him.
    Leo: When they just first started, I noticed that they were throwing the ball at like a high and back and forth pace. So that’s like what it is. And then right here is when they just started like doing it like midway. And then once the guy who was over here drops it completely, picks it up and then drops it again.

    +++ 00:05:33 +++
    Student: They start with the ball in the middle of the air because it’s got to start at zero.
    Student: I think they’re already at a certain height--
    Leo: Yeah.
    Student: -- because it wasn’t their hand and it’s not put on the ground. So it’s already at a certain height.
    Krista McAtee: With this group, to get them to say well, I don’t understand why this or wouldn’t your graph-- shouldn’t your graph look like this is a huge risk for them and to be able to hear that in a positive way and be accepting of that feedback is a big thing.

    +++ 00:06:00 +++
    Krista McAtee: Can you think of something if we graphed it again that you might add to your graph?
    Student: Where he had dropped it and then, tied his shoe, and then dropped it again.
    Krista McAtee: Okay. Okay. So what I’d like you to do is talk to your partners about how you could improve your graph.
    Student: I don’t know how many times they threw it back and forth but they were just throwing it. When he picks it up, drops it, picks it up again, drops it again--

    +++ 00:06:22 +++
    Krista McAtee: The thing that they have to break through, that all students have to break through, is they have to learn that they have to persevere. They have to keep trying. They have to take a risk and they have to be okay with that they’re going to mess up and they can try it again and they might mess up again and that’s okay at failing. It’s a huge piece of it, that they’re going to fail at times and that’s okay.
    Leo: Threw the ball.
    Krista McAtee: Yeah. So he threw it, it went up, right?
    Leo: Yeah.
    Krista McAtee: And then once it got to the highest point that it went to, where did it go?
    Student: It would go down.

    +++ 00:06:49 +++
    Krista McAtee: It would go down again.
    Student: See?
    Leo: Oh.
    Krista McAtee: Is that what you’re thinking?
    Student: Yeah.
    Krista McAtee: Okay.
    Student: Keep it. You keep going like this.
    Student: No, he throws it a little bit higher than the first one.

    +++ 00:06:58 +++
    Krista McAtee: I saw a huge learning curve. I saw things that they didn’t understand in the beginning that they were able to articulate and even ask each other about later on in the lesson. I think that they understood the idea of heighth over time. I think that they understood the idea that the ball went down to the X-axis and it was at zero. Those were big concepts that I wanted them to get.
    Krista McAtee: What did you notice that was similar in the graphs and what was different?

    +++ 00:07:27 +++
    Student: They start at the middle at the-- like right there and our side’s already at the bottom at the zero.
    Krista McAtee: Okay. And why did you guys choose to start yours at zero?
    Student: Because when they’re starting, the ball’s in the glove at zero. It’s not high or anywhere. It’s just there.
    Krista McAtee: So you’re thinking about zero heighth as this high?
    Student: Yeah.
    Krista McAtee: All right. I feel like everybody is capable of getting there. they have different ways to think about whatever they’re approaching and I think that it’s really important to validate those different ways.

    +++ 00:08:01 +++
    Student: When they start, they start with the glove and that’s zero for them. So when it misses the glove and it goes under where they started, is that negative?
    Krista McAtee: What do you guys think about that?
    Student: Well, I already know. I thought it about. I know it’s wrong already.
    Krista McAtee: Well, no one’s saying it’s wrong but he’s wondering, right--
    Student: No, I know what he’s saying.
    Krista McAtee: Okay. Would you change something?
    Student: Yes.
    Student: Yes, we would.
    Krista McAtee: What would you change?
    Student: The height where it starts.
    Student: The height where it starts.
    Krista McAtee: Okay. So you could change where it starts or you could start that at-- you could say this is zero and--
    Student: Then you just put a negative.
    Krista McAtee: And then have that go down below the zero.
    Student: Yeah.

    +++ 00:08:33 +++
    Krista McAtee: My goal is that they feel like they have some math skills--
    Krista McAtee: -- that they can think about math, that there’s not just one right way to answer any or approach a math problem, but that they have something that they can contribute and that they are mathematical thinkers.
    Student: There you go. That makes sense.
    Tch Teaching Channel
    #### End of C0804_001006_Sonoma_Class_FINAL_SD.mp4 ####

School Details

Sonoma Valley High School
20000 Broadway
Sonoma CA 95476
Population: 1312

Data Provided By:



Krista McAtee
Jon Southam
Chris Anspach



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Lesson Idea

Grades 9-12, All Subjects, Class Culture

Lesson Idea

Grades 9-12, ELA, Class Culture

Teaching Practice

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