Series: Collaborating to Develop Mathematical Ideas

Math.6.RP.A.2

Common core State Standards

  • Math:  Math
  • 6:  Grade 6
  • RP:  Ratios & Proportional Relationships
  • A:  Understand ratio concepts and use ratio reasoning to solve problems
  • 2: 
    Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to 0, and use rate language in the context of a ratio relationship. For example, \"This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.\" \"We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.\"


    Expectations for unit rates in this grade are limited to non-complex fractions.

Download Common Core State Standards (PDF 1.2 MB)

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Math.6.RP.A.3

Common core State Standards

  • Math:  Math
  • 6:  Grade 6
  • RP:  Ratios & Proportional Relationships
  • A:  Understand ratio concepts and use ratio reasoning to solve problems
  • 3: 
    Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.


    a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
    <br />
    b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?


    c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.


    d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Download Common Core State Standards (PDF 1.2 MB)

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Math.7.RP.A.2

Common core State Standards

  • Math:  Math
  • 7:  Grade 7
  • RP:  Ratios & Proportional Relationships
  • A:  Analyze proportional relationships and use them to solve real-world and mathematical problems
  • 2: 
    Recognize and represent proportional relationships between quantities.
    <br />
    a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.


    b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.


    c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
    <br />
    d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Download Common Core State Standards (PDF 1.2 MB)

Reflecting on Teaching & Learning About Ratios
Lesson Objective: Reflect on lessons about ratios and proportional relationships
Grades 6-8 / Math / Teacher Collaboration
Math.6.RP.A.2 | Math.6.RP.A.3 | Math.7.RP.A.2

Thought starters

  1. How did the lesson change depending on each classroom context?
  2. What do the teachers learn from looking at student work?
  3. How do the teachers support each other and push each other's thinking?
10 Comments
Great way to enhance lessons when the strategies for teaching are based on student work. Collaborating with other teachers on a common lesson and shared experiences definitely help develop a richer lesson delivered by the teacher and a deeper understanding and potential growth for the individual student.
Recommended (1)
Thanks Joeleen. The growth I had from this conversation has helped deeper my own understanding of math. It is amazing to know how much I learn everyday.
Recommended (0)
I just previously watched your video on Purple Paint I and this is a great follow up. It is evident that looking at student work and collaborating amongst grade levels is very valuable. It allows you to see what you need to teach in more depth, and the other teachers to see where their students had began with proportional reasoning.
Recommended (0)
Sharon... thank you for your comment. I agree. Seeing task vertically aligned and thinking about how they progress over time is very helpful. Thanks for watching!
Recommended (0)
I always think, when watching these videos about collaboration, how much time is spent at collaboration and how little time teachers actually have to plan, prep, etc. and it can't all fit! But I just experienced an Aha moment realizing that the up front time load is far greater than continuing time load, year to year. Next year when this team revisits this lesson, they will stand on the work they did previously, and the time required will be far shorter in ensuing years. A good argument for stability in the workplace!
Recommended (0)

Transcripts

  • Reflecting on Teaching & Learning: Ratios & Proportional Relationships Transcript

    +++ 00:00:04 +++
    Crystal Morey: I really am happy that we

    Reflecting on Teaching & Learning: Ratios & Proportional Relationships Transcript

    +++ 00:00:04 +++
    Crystal Morey: I really am happy that we get to come together, talk about student work and really kinda look at where we're going in our own math knowledge and where we go from here.

    +++ 00:00:12 +++
    Crystal Morey: As an educator, I can only grow my students as much as I grow myself, and only through by collaboration by looking at student work and challenging my own thinking as a teacher can I grow.
    Card:
    Illustrative Mathematics:
    Reflecting on Teaching & Learning
    Ratios & Proportional Relationships

    +++ 00:00:30 +++
    Crystal Morey: So I know that we just got done teaching similar lessons together.
    Card:
    Crystal Morey's 6th Grade Math Class
    Enumclaw Middle School, Enumclaw, WA

    +++ 00:00:34 +++
    Crystal Morey: Perfect purple paint is made by taking two cups of blue paint, okay, and putting it together, and taking three cups of red paint and putting it together. That makes one batch of perfect purple paint.
    Lower Third:
    Crystal Morey
    Middle School Math Teacher
    Enumclaw Middle School, Enumclaw, WA

    +++ 00:00:52 +++
    Crystal Morey: In today's lesson, we were starting to look at how do we use fractions and apply that to our ratio understanding.
    Lower Third:
    Jana Dean
    6th Grade Math & Science Teacher
    Jefferson Middle School, Olympia, WA

    +++ 00:00:59 +++
    Jana Dean: This morning, Wendy and Crystal and I sat down together and first, we came to a better understanding of how each of us had presented the task and facilitated it with our students, and then we took a look at students' work.
    Card:
    6TH GRADE TASK
    Perfect Purple Paint is made by mixing 2 cups blue paint to 3 cups red paint

    +++ 00:01:13 +++
    Wendy Hughes: Just kind of a natural progression. They started ultimately to make tape diagrams. They don't even know what a tape diagram is. They don't have that background knowledge, but--
    Lower Third:
    Wendy Hughes
    6th Grade Math & Science Teacher
    Jefferson Middle School, Olympia, WA

    +++ 00:01:24 +++
    Wendy Hughes: When we got to look at one another's student work, we got to see what kids were doing at the next step or the step behind, and figure out where as teachers we would go next and what would make sense for where the kids are and where we wanted 'em to go.

    +++ 00:01:39 +++
    Wendy Hughes: You'll see the beginning of the tape diagram and then you see it growing and growing into 20 cups.

    +++ 00:01:45 +++
    Crystal Morey: So that connects, I think, to this student who said "Why would I-- all the other kids were using 20 blocks--"
    Wendy Hughes: Uh-huh.

    +++ 00:01:50 +++
    Crystal Morey: And he said "Why would I use 20 blocks? I'll just create a key." And this key'll say one block equals four. And so, naturally, right, he was creating this tape diagram, but he was just trying to really look for a simple way to show his thinking.

    +++ 00:02:03 +++
    Crystal Morey: Only by looking at student work could we see that there was repeated connections through our students' work. Yet I also got the opportunity to see how Jana distinguishes between equivalency and ratio.
    Card:
    7TH GRADE TASK
    Perfect Purple Paint is made by mixing 1/3 cup blue paint to 1/2 cup red paint.

    +++ 00:02:13 +++
    Jana Dean: About half my class, and maybe it just started somewhere and they started to see "Whoa, that's really elegant," but about half my class demonstrated that they were able to set up the two fractions and compare them. I have half and I have a third, so I need to represent them as sixths. So here's my half, here's my third.
    Card:
    Jana Dean's Math & Science Class
    Jefferson Middle School, Olympia Washington

    +++ 00:02:35 +++
    Jana Dean: And then on the floor, erasing the white board marker, they just get rid of everything that doesn't represent the ratio. All of this goes away, we don't have to worry about it, and then I can just use a tape diagram and scale it up, but put exactly the same number in each box so that I have 20 cups.

    +++ 00:02:52 +++
    Crystal Morey: And I think this is where I need to go, so as I really look back, my kids-- I don't think right now I've given them access to this linear model in this particular way, because many of my students were setting up tape diagrams with the three connecting to the blue, and the two connecting to the red, just based solely on the denominator.
    Jana Dean: Right.

    +++ 00:03:11 +++
    Crystal Morey: And though they were getting 12 and eight, the 12 and eight were flipped.
    Jana Dean: Flipped.

    +++ 00:03:14 +++
    Crystal Morey: But just what you said, you said they could also compare a half to a third.
    Jana Dean: Right.

    +++ 00:03:19 +++
    Crystal Morey: That's the skill that some of my students did not have access to--
    Jana Dean: Right.

    +++ 00:03:23 +++
    Crystal Morey: -- and conceptually weren't understanding. This student originally came up with this strategy--
    Card:
    6TH GRADE TASK
    Perfect Purple Paint is made by mixing 2 cups blue paint to 3 cups red paint.

    +++ 00:03:28 +++
    Crystal Morey: -- to show three to two, and it was kind of all over. So initially this student had 12 to eight cups. That made sense. And then the student was convinced in their thinking by other students, that it needed to be 10 to 10; that even though the initial ratio wasn't equal, that the ratio should now be equal. And as that student made that turn from the correct answer to the incorrect answer--
    Jana Dean:
    Wendy Hughes:

    +++ 00:03:52 +++
    Crystal Morey: -- I find that's one of my hardest times as an educator to do what you were saying, Jana, which was to just kind of let it be for a second and see how it plays out and not jump in and rescue them so quickly.

    +++ 00:04:02 +++
    Jana Dean: Good teaching has very little teacher at the centre, teacher in the middle of things, teacher delivery; and a whole lot of student thinking and student communicating, because you're listening for student thinking all the time, and you create opportunities for students to express themselves in ways that make sense to them.

    +++ 00:04:23 +++
    Crystal Morey: Eventually, a few minutes down the road, as I came back around, the student, he went back to their original thinking and could articulate fairly clearly why, and convince some of the other-- other students were now making models that represented that reasoning.

    +++ 00:04:37 +++
    Crystal Morey: I think Jana, Wendy and I, all three of our classrooms, we were all looking at making sense of the students' thinking right where they're at. Students walk in our door, we meet them exactly where they are, and we move them forward, and we do that with a variety of different strategies. More importantly, though, we let our students teach us and teach one another every single day. We facilitate that very carefully.

    +++ 00:05:00 +++
    Jana Dean: So he started out in the wrong direction. I didn’t correct him, because--

    +++ 00:05:06 +++
    Wendy Hughes: But the nice thing is he realized he was going the wrong way and turned around.

    Jana Dean: It wasn't really wrong; it was just not a straight line.
    Wendy Hughes: Right. Exactly.
    Jana Dean: That's what problem-solving is all about.

    +++ 00:05:14 +++
    Crystal Morey: I saw this misconception too; that they read as "20 cups of each," not 20 cups together, and that understanding of part-part-total was--
    Wendy Hughes: Why they went to 10 and 10.
    Crystal Morey: Right. Why they went to 10 and 10. He was making 20 cups of each, and seeing that connection.

    +++ 00:05:29 +++
    Jana Dean: Collaborating around student work, observations, means that I might see something in it, Wendy might see something else in it, Crystal will see a third thing in it, and then our conversation is far richer.
    Card:
    7TH GRADE TASK
    Perfect Purple Paint is made by mixing 1/3 cup blue paint to 1/2 cup red paint.

    +++ 00:05:45 +++
    Crystal Morey: This student strategy, he was having trouble articulating it, but he started by saying he knew that it had to be out of six, 'cause he knew he just needed to find something that they both fit into. So he said three and two, and so he built this model, and he said "This represents five-six." And then this student, he made a bunch of this five-six, like this. He just started making 'em. But then he said "I want to get to whole cups,"
    +++ 00:06:11 +++
    so he started off by building two of these models just like this.
    Jana Dean: Okay, so what if I doubled my recipe.

    +++ 00:06:15 +++
    Crystal Morey: And he goes "Yeah, but I want to get to whole cups, and then I'll figure out how much is left over." So he said "I'm gonna get to whole cups, and I have one whole cup, and two thirds left over."
    Jana Dean: Right.

    +++ 00:06:30 +++
    Crystal Morey: So he said "In two batches there was one and two-thirds."
    Jana Dean: Right.
    Crystal Morey: And so then he started making--
    Jana Dean: There it is. It's right here.

    +++ 00:06:38 +++
    Crystal Morey: "So then when I kept building these, I noticed that I got to five."

    +++ 00:06:43 +++
    Wendy Hughes: I haven't in the past had the opportunity to share the work that they do with another teacher in the same way, in a middle school setting, and it's amazingly powerful. We just bring the sort of neutral perspective that I think you don't always get when you're looking at your own student work, but when you're looking at someone else's you get to see the math and not the kiddo.

    +++ 00:07:08 +++
    Jana Dean: Whenever you have people who are all able to focus on the same thing at the same time, then amazing things happen between people, and that's what education is all about anyway.

    +++ 00:07:16 +++
    Crystal Morey: So it seems like in all of this, that if we don't really kind of get messy with the math ourselves, that it's really hard for us to come back and look at student work. So some way or another we have to have the opportunity to really do these problems first a bit, learn all the different strategies as learners, before we come back as teachers.

    +++ 00:07:36 +++
    Crystal Morey: So often we work on instructional strategies, developing assessments, and yet we forget to go back to what do we know about math? What are our own weaknesses in math? And what does our students' work tell us? As an educator I can only grow my students as much as I grow myself.

    +++ 00:07:52 +++
    Crystal Morey: Well, I want to thank you guys for really coming and talking about the students' work. It's so nice to be able to see this work connected to this work, and what misconceptions your student had that my student also had, that same misconception. And also, Jana, to see how I need to continue to grow in my own practice, because I think I need to continuously get better and give my students access to strategies that are working, with other kids' conceptual development. So, thank you.

    +++ 00:08:17 +++
    Jana Dean: You're welcome. Yeah, I don't have those blocks.

    Jana Dean: Do you have any?
    Wendy Hughes: No, I don't have any.
    Crystal Morey: I just bought 'em.
    #### End of C08004_001001_ENUM_COLLAB_FINAL.mp4 ####

School Details

Enumclaw Middle School
550 Semanski St
Enumclaw WA 98022
Population: 448

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Teachers

Crystal Morey
Jana Dean
Wendy L. Hughes

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