Series Engaging Students with "Productive Struggle": Understanding Proportional Relationships


Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP1:  Make sense of problems and persevere in solving them.

    Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, \"Does this make sense?\" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Download Common Core State Standards (PDF 1.2 MB)


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)

Understanding Proportional Relationships

Lesson Objective: Recognize proportional relationships between two quantities
Grade 7 / Math / Assessment
Math.Practice.MP1 | Math.7.RP.A.2


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Discussion and Supporting Materials

Thought starters

  1. How does Ms. Walker encourage her students to take responsibility for their learning?
  2. Why does Ms. Walker use feedback questions instead of comments when responding to student work?
  3. What is powerful about grade level teachers looking at student work together?


  • Private message to Henrik Glensbo
I look at your videos often and find them very inspiring. Could anybody enlighten me - there is a figure District Spending - what does this amount cover? Thanks in advance
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  • Private message to Marty Joyce
Awesome lesson and enthusias by the teacher. I love the accessibility and structure. Noting the difference between being allowed to toast in multiples of 2 versus odd amounts of slices of toasts. I will definitely consider teaching this lesson as a replacement unit for this topic in CPM. Thank you.
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  • Private message to Kathy Helgeson
To: Karen Black- Many of our schools in Oregon are using EngageNY. It is a free, online, Common Core aligned curriculum. It was created by NY with Race to the Top funds, and is now accessible to all. It provides a great deal of explanation about how the math should be taught. You can access the curriculum at Eureka Math sells the printed curriculum, but we were able to send it to Staples and have it bound into modules so teachers would have a hard copy in addition to the online version. Good Luck
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  • Private message to Karen Black
I noticed in the supporting materials that your lessons come from MAP which supports grades 6-high school. I teach 5th grade math. Are there resources that elementary teachers in your district use to support their teaching? The math series we currently have does not support common core, our district has no monies to get us new materials, so we are on our own to find quality resources.
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  • Understanding Proportional Relationships Transcript

    Speaker 1: Today, we're going to do another activity where we're talking about proportional reasoning, and the

    Understanding Proportional Relationships Transcript

    Speaker 1: Today, we're going to do another activity where we're talking about proportional reasoning, and the lesson we're going to do today has to do with looking at some stacks of paper. Okay?

    My name is Terry Walker. I teach at Twenhofel Middle School in Kenton County, Kentucky. I teach seventh-grade math.

    So, this is five centimeters, and this is 500 sheets of paper. And what we're going to do today is we're going to see how many pieces of paper there are going to be in your stack without counting it. Okay? I'm just randomly going to give you some paper - you and your partner.

    We're trying to approach math in a way that is different than how many of us learned it; that is much more logical; much more based on understanding, as opposed to procedure.

    Speaker 2: A hundred divided by 10 for each of these millimeters will be ten sheets of paper per [millimeter?].

    Speaker 1: Ooh. So, you were going down to the millimeter level.

    Speaker 2: Yeah.

    Speaker 1: Okay.

    We have gone from a teacher-centered classroom where the teacher is doing all the work and just giving kids information to the kids doing the math and the kids acquiring math skills and thinking skills that they wouldn't've gotten when they were listening to me talk.

    Speaker 3: So, I got 100 and then [6?] [?].

    Speaker 1: What does that 100 tell you?

    Speaker 3: That 1 centimeter is 100.

    Speaker 1: Okay. And then what did you do?

    Speaker 3: Divided that by 2, because it's only half a centimeter, and I got 50.

    Speaker 4: So, 7 times - so, it would .7. So, try .7 times 100?

    Speaker 1: My question is, is it a proportional? Is the number of sheets of paper proportional to the height of the stack, or vice versa? Are those two quantities in a proportional relationship?

    Speaker 4: Yes, because every centimeter is 100 -

    Speaker 5: Sheets of paper.

    Speaker 4: - yeah, sheets of paper. And then if you put that into a graph, it would be linear, so it'd be - and it also has a constant ratio of 1 centimeter to 100 sheets of paper.

    Speaker 1: You said a lot right there. Wow. That was a lot - wasn't it? You said "constant ratio." You said, "If we graph it." Did you graph it?

    Speaker 4: No.

    Speaker 1: No? If I graphed it, what's going to happen?

    Speaker 4: It's going to be in a straight line, like a linear graph.

    Speaker 1: So, kind of like that. Okay, awesome.

    I feel that school mathematics has a role [into?] something that's really hard to recognize as mathematics, because the students are involved in learning tiny, little fragments of knowledge. The only way to break out of that trap is by making the mathematics meaningful.

    Okay. So, go ahead and keep your papers up to the side. We're going to start a pre-assessment for our next lesson.

    So, a formative assessment lesson is a type of lesson that we in our county were fortunate enough to stat using when we first started looking at adopting the Common Core. They are lessons in which we really look at what kids know and what they're thinking, and then we go back and assess what they've learned and their growth to kind of form where we're going next and what their misconceptions still are.

    We've been talking for about a week and-a-half about proportional relationships, about quantities that are in a proportional relationship. So, what we're going to move on to in the next couple of days is a formative assessment lesson. We've done these before. We're going to kind of see what our level of understanding is.

    So, I hand out a pre-assessment, and I tell the kids, "We're going to have a lesson that's going to help you improve." This assessment - and, of course, the students always want to know, "Am I getting graded on it?" And I like saying to them, "No. You're not getting graded," and you can see them relax about it and stat saying, "Well, let me focus on what this is asking me to do."

    Speaker [6?]: The formative assessment lesson, proportional and non-proportional situations, is designed to get out onto the table - kind of expose - any of the misconceptions, gaps or errors that the kids have.

    Speaker 1: As I look at their misconceptions and we go through the formative assessment over the next, you know, two class periods, we can get some of those misconceptions looked at through classroom discourse, through them talking with their partner, and through people working together in the classroom on the misconceptions.

    Okay. So, I'm going to ask everyone to stop as far as you go. That's okay. I want you to turn your paper over and make sure you write your name at the top, because I want to be able to know whose work is whose. And then just pass them forward for me, please.

    The beauty of the formative assessment lessons is that I collaborate with my colleagues. So, we all made it so that we're giving the pre-assessment and we're teaching it on the same days, so we can meet around student work. Do all of our kids have similar misconceptions? What are the most important ones? What misconceptions are important for us to get rid of through the lesson, and how are we going to do it?

    Thanks, guys. Okay. So, the first thing we're going to look at is identifying the math of the lesson so that we know, you know, what to look for when we're looking for misconceptions.

    So, one of the things that we've really tried to do is make sure that teachers have a lot of opportunities to collaborate with one another. One of the things that we've found most important is making sure that teachers collaborate at course-specific levels. We're trying to get very focused on the math at their grade level and how that impacts their classroom.

    Either they got it, or not, Denise. So, how can you word that so a student doesn't scale the -

    Speaker 7: Well, how 'bout they're not looking at the fraction as one quantity? I think they're just misusing the ratio in this.

    Speaker 8: Yeah, I'd say that. Just - yeah, I think that's right.

    Speaker 9: Misusing it.

    Speaker 10: Misusing [?].

    Speaker 8: Misusing the ratio.

    Speaker 1: We try to identify common misconceptions among the students. Then what they do is they look at the student work for a class, and they identify which students have those misconceptions, and they can use that data to help group[ed?] students for the task the next day - for the collaborative activity.

    Speaker [1?]: The feedback is designed to cause students to think more deeply about the mathematics, and I think we're kind of asking them to loo for a pattern and apply it to a different situation with a more difficult number.

    Speaker [?]: Right.

    Speaker 1: Then they work together as a group to develop feedback questions to be used during the lesson over the next two days.

    Speaker 9: You asked what process would you use. And then could you use the same process for 17-1/2?

    Speaker 1: Basically, what the questions do is they try and address a misconception that was seen in the student work and basically find out a little bit more about what the student is thinking about that.

    I kind of like saying, "What is the relationship between red cans and blue cans. Can you show this in a table?"

    Speaker [?]: Right.

    Speaker [?]: Oh.

    Speaker [?]: Showing this in a table.

    Speaker 1: Or, show this with - and then we could put "table," "list" - whatever.

    Speaker [?]: Or, "model" - yeah. [Crosstalk] -

    Speaker 1: Yeah, "table" or "model."

    Speaker [?]: They could draw it.

    Speaker 1: I like that. And we want to make sure it's not just yes or no. We want to make sure that we're really learning something else about student understanding.

    Speaker [?]: Like that question about, "How many cans of blue would you use for one, single, red can?"

    Speaker 1: And [crosstalk] unit.

    Speaker [?]: Getting to the unit - right.

    Speaker 1: Because once they get that, the fraction doesn't matter anymore.

    Speaker [?]: Right.

    Speaker 1: This process really gives teachers the same type of learning environment that we're giving to students. So, we want students to have the productive struggle. We want students to be able to work through their misconceptions and not just to be told by a teacher, "This is what you need to do, and why." And so we're trying to recreate that experience for teachers as well, because that's how real learning works.

School Details

Twenhofel Middle School
11846 Taylor Mill Road
Independence KY 41051
Population: 816

Data Provided By:



Teri Walker
Jenny Barrett
Ann Shannon


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