Series: AFT CCSS Math

Math.Practice.MP2

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP2:  Reason abstractly and quantitatively.


    Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Download Common Core State Standards (PDF 1.2 MB)

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Math.Practice.MP3

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP3:  Construct viable arguments and critique the reasoning of others.

    Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Download Common Core State Standards (PDF 1.2 MB)

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Math.8.F.B.4

Common core State Standards

  • Math:  Math
  • 8:  8th Grade
  • F:  Functions:
  • B:  Use functions to model relationships between quantities.
  • 4:  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Download Common Core State Standards (PDF 1.2 MB)

Conjecturing About Functions
Lesson Objective: Analyze patterns and represent functions
Grades 6-8 / Math / Reasoning
Math.Practice.MP2 | Math.Practice.MP3 | Math.8.F.B.4

Thought starters

  1. How do students represent functions in multiple ways?
  2. How does noticing structure prepare students for making conjectures?
  3. What can you learn from Ms. McPhillips about using the Common Core with students?
31 Comments
Thanks, Ms. McPhillips -- this is useful to me for modeling how to encourage and probe students' thinking, and how to debrief such activities. I will share it with my 6-8 math department at a math meeting and we can all brainstorm how we can apply your techniques to our lessons. I also want to directly steal your idea of the closure piece of "which math practices do you think you used today - discuss" and perhaps make the very same bulletin board display in my room, neither of which I had considered before watching your class on video.
Recommended (1)
Excellent video.....this will be so helpful to me. I will definintely use this lesson next year. I also love the idea of having the students discuss which mathematical practice they used. Where can I get those "mathematical practice" posters you had on your bulletin board? I will share this video with my math department ...thank you
Recommended (2)
I agree with Kathleen and Lisa. This was a very good video. I also have the same question as Kathleen: Where can I get those posters?
Recommended (1)
Thank you everyone! I am glad the video is useful to you. The students really enjoyed the process of being video taped, and I was so proud of the great thinking and reasoning the engaged in during this series of lessons. Jordan School District in Utah created the posters - which seem to be posted all over on different internet sites. This site in particular has the posters at different grade bands. I hope you find them useful - I know that I did! http://teachertreasures.com/CCSS_Math_Posters.html Audra McPhillips :)
Recommended (2)
Thank you so much Audra. I was so amazed with how well the students knew what mathematical practice they were using. I definitely want to incorporate this in my lessons.
Recommended (1)

Transcripts

  • Conjecturing About Functions Transcript
    Luna Productions
    Program: Audra_function
    Teacher: Audra McPhillips

    Voice identification: Teacher
    Student (for any/all student who

    Conjecturing About Functions Transcript
    Luna Productions
    Program: Audra_function
    Teacher: Audra McPhillips

    Voice identification: Teacher
    Student (for any/all student who speaks)

    My name is Audra McPhillips, I'm a math coach for the West ___ public schools and I taught an 8th grade lesson at Dearing Middle School today on conjecturing above functions.

    If you remember yesterday we were looking at a set of different functions and we were trying to come up with different conjectures ….

    Teacher: The heart of the lesson, 8th graders are expected to be able to not only understand what functions are but be able to represent them in variety of ways.

    Teacher: For a couple of days before this lesson the students had examined a particular set of functions.

    Teacher: So we started the lesson today by having them share their thinking with the class

    Student: So we proved that by making ….like making a picture of the dots

    Teacher: What up there convinces you?

    Student: Oh, well, there’s the dot in the middle that stands for the extra 1 and then ….

    Teacher: I think more importantly this lesson and this series of lessons is really getting them to move beyond the set of problems that’s provided to start to notice important math structures and to really make a conjecture.

    Student: I think it should be the number of minutes and the Y should be the total

    Teacher: When they get to the point where they’re making a conjecture they’re really engaging in deep math reasoning, which is what the common core is looking for.

    Teacher: I want you to take a look at these three different patterns, OK? There are three of them. There’s pattern A, pattern B, and pattern C.

    Teacher: I chose the three functions that we used for this lesson because all three of them had the same rate of change. That’s a structure, that’s a pattern that I want them to recognize. I also chose those three because they all had a different Y intercept. They all started in a different place and I knew that students would struggle with that a little bit because they hadn’t had a chance to think about what happens when the beginning is not clear.

    Teacher: So I first asked them to take some individual time to look at that set of functions and just see what they noticed.

    Teacher: What did you notice about this set before we get started solving them? Matthew?

    Student: They all start at different times.

    Teacher: They all start at different times. Tell me more about that.

    Student: Like one will start at the beginning and then one will start at one minute and then [noise] one doesn’t have anything at all

    Teacher: That’s good stuff, Matthew. Chloe, what’d you notice?

    Student: In pattern A, at the beginning, they didn’t ….at one minute they had one dot in each arm and pattern B at one minute they had two dots in each arm, and in pattern C they had no arms at one minute

    Teacher: And Michael, one more. Do you want to add to that?

    Student: They all have the same number of arms.

    Teacher: They all have the same number of arms. Ohhhhhhhh

    Teacher: In this particular case there's a lot of reasoning going on. They’re looking for repeated reasoning, they’re looking for that structure, they're reasoning abstractly and quantitatively. All of those reasoning math practices that are difficult to get in kind of fall together when you have them work on conjectures.

    Teacher: I want you to try to make a conjecture. So the end goal is to kind of analyze this whole set of functions and think about is there a conjecture you can make. Can you think of a math statement you can make that kind of goes beyond this set that will always hold true, and remember some of the things we talked about that make a strong conjecture. So if you think you’re onto something jot some notes in your conjecture paper while you’re finishing up your work and we’ll see if anybody kind of gets there today.

    Student: So first of all we need a conclusion ….

    Student: Because to get Y we need the equation ….

    Teacher: Once you get your graph done color code everything so it matches so the other people will be able to understand your thinking, OK? That was smart to go there first, I like it.

    Teacher: Well I definitely really steered students towards using color. Students need to understand this algebra but they also need to kind of have that picture in their mind to back it up. So the color really [weird] helps them to kind of develop that conceptual understanding of how all of those representations fit together so they’ll be able to apply that later on.

    Teacher: We know one minute, two minutes, three minutes, what do we not know that you want to know? We don’t the beginning, right. So let’s do that

    Teacher: Mike, Tony and Eric, they were struggling with that last growing pattern. It was difficult for them because there was no beginning provided, they were going to have to kind of think about a beginning and that beginning was going to get into some negative numbers.

    Teacher: So from here to here how much do we grow?

    Student: So there’d be like negative three dots

    Teacher: Ohhh, where’d that come from, that was pretty impressive. Go ahead, tell me what you’re thinking, Michael

    Student: If you would subtract 4 from what ___ you have a negative 3

    Teacher: Right, so tell them why though, because I don’t know that they’re following that. So what did you notice when I said what happens from here to here?

    Student: That it adds 4

    Teacher: Adds 4, adds 4, adds 4. So you’re saying if we go this way we’re adding 4 so ….

    Student: …if we go this way you’re subtracting 4

    Teacher: So to get from 9 to 5 you ….

    Student: Subtract 4

    Teacher: To get from 5 to 1 you ….

    Student: So he’s saying that the beginning must have …

    Student: A negative three dots ….

    Teacher: Which is kind of weird to think about but I think mathematically it makes a lot of sense. Does that help ….

    Teacher: He was recognizing that inverse which is really powerful. He was able to justify his thinking and explain it to the rest of the group and just another ah-ha moment that was kind of exciting.

    Teacher: Shirley! I love it, say that again, say that to Evan.

    Student: It’s the same rate for each one …

    Teacher: So each one has the same rate of change and what’s the rate of change?

    Student: Four

    Teacher: Four. And so what does that mean for every single one of these functions, the formula is always going to start what?

    Student: Uh, Y equals 4 X

    Teacher: Y equals 4 X. Awesome.

    Teacher: That was a really important structure for groups to notice and it was exciting to hear them kind of articulate that and have that big ah-ha moment which was going to support all of their other work with all of the other representations.

    Teacher: What were you noticing?

    Student: Equations are the same rate of change but different initial values will make parallel lines because ……

    Teacher: Zachary and Steven were kind of the first group that had gotten beyond the problem set at hand, gotten beyond solving and representing.

    Teacher: Equations that have the same rate of change, so tell me about that. These all have the same rate of change?

    Student: Um yeah, this one goes up by 4, this one goes up by 4

    Teacher: They had graphed two of the functions on the same plane and I could hear them talking about the fact that they were parallel so I immediately jumped into that conversation because I was kind of excited that they were noticing that phenomenon.

    Teacher: They’re all going to end up being parallel lines and you’ve only graphed two of them so what do you think? This is your 4 X, this is ….thank you. This is 4 X plus 5, 4 X plus 1. So where do you think the other one might fall?

    Student: Probably like to the right

    Teacher: They were really starting to notice an important math structure that went beyond the set of problems and they were really starting to make a conjecture.

    Teacher: I’ve noticed some people already starting to notice some important math structures, some things that might get to the conjecture point. I don’t think we’re going to get to the point where you can all make a very solid conjecture that we can try to reason about. I think we’ll save that for tomorrow where we kind of get to the final conjecture

    Teacher: At the end of the lesson after students had worked on the problems they worked on an exit slip for me so I could see where their thinking was at the end of the lesson, and then I asked them what they noticed and I was excited with some of the results.

    Teacher: If you had to pick two or three math practices that you think you engaged in the most today, that were the most helpful to you while you did this work, go ahead and chat with each other ….

    Teacher: . I think it's really important with the common core standards since the standards for mathematical practice are tools for the students. These are tools that they need to kind of hone throughout their K-12 experience. If they can do these eight things, they're going to be successful with any kinds of problems. So I want them to really stop and think about their own thinking, what strategies were you using, which math practices were you engaging in that allowed you to be successful with this lesson today.

    Teacher: Michael’s group, I heard you guys talking about something good. What was it?

    Student: Look for and make use of structure

    Teacher: So go ahead and find the yellow one guys, look for and make use of structure. So Michael, why was structure important in this lesson? What was your group discussing?

    Student: We were discussing on how to find the beginning and um ….like constant rate of change for each problem

    Teacher: So Michael is saying we had to look for important math structures while we were building this function. We had to look for things like rate of change and Y intercept and beginning, and so he said they were examining all those structures to kind of make connections. So yeah, I think people who make good conjectures use structure really well.

    Teacher: This is the kind of lesson that generally takes two days. They’re solving these problems but you know they're solving them as a tool to kind of go beyond and they were already getting to that point today which was good because that means tomorrow they’ll be able to test it with a wide range of items, continue to color code their work so other people can understand it and most importantly really try to get down to the math of why is this conjecture seeming to be true.

    ² end of transcript

School Details

John F. Deering Middle School
2 Webster Knight Dr
West Warwick RI 02893
Population: 994

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Audra McPhillips

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