Series Engaging Students with "Productive Struggle": Deepening Understanding: Linear Equations


Common core State Standards

  • Math:  Math
  • A:  Algebra
  • CED:  Creating Equations
  • 2: 
    Create equations in two or more variables to represent relationships
    between quantities; graph equations on coordinate axes with labels
    and scales.

Download Common Core State Standards (PDF 1.2 MB)


Common core State Standards

  • Math:  Math
  • A:  Algebra
  • REI:  Reasoning with Equations and Inequalities
  • 1: 
    Explain each step in solving a simple equation as following from the
    equality of numbers asserted at the previous step, starting from the
    assumption that the original equation has a solution. Construct a
    viable argument to justify a solution method.

Download Common Core State Standards (PDF 1.2 MB)


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)

Deepening Understanding: Linear Equations

Lesson Objective: Assess understanding of linear equations
Grades 8-12 / Math / Assessment
Math.A.CED.2 | Math.A.REI.1 | Math.Practice.MP2


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Discussion and Supporting Materials

Thought starters

  1. What do the teachers discover during their review of the post assessment?
  2. How does formative assessment inform future instruction?
  3. Why does Ms. Morehead ask her students to make predictions?


  • Private message to Megan Wooldridge
I'm an instructional coach for a relatively small middle school, and we're trying to get our teachers and students to engage in this type of dialog. How long did it take you to train your students to engage in productive struggle? How many of these types of lessons do you do in a quarter? And finally, how many days did it take to complete this FAL?
Recommended (0)
  • Private message to roger connor
I like how the staff got together to discuss testing. I am at a very small school with only 12 teachers. We do get together but we all teach something different.
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  • Private message to Gretchen Vierstra
Hi Blake, this lesson video is one of three about this entire process, which takes place over three class periods. Take a look at the related blog post for the other videos and more details:
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  • Private message to Blake Boffa
Were the pre assessment and post assessment given on the same day? How can you expect student growth after one period? Or was the pre assessment given previously and did you use those grades to drive instruction leading up to the post assessment?
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  • Private message to Gretchen Vierstra
Hi Jodi, the lesson is under high school on the MAP site. It's Solving Linear Equations in Two Variables. Hope that helps!
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  • Deepening Understanding: Linear Equations Transcript

    Speaker 1: I like how she used multiple methods. She used the tables, the graphs -


    Deepening Understanding: Linear Equations Transcript

    Speaker 1: I like how she used multiple methods. She used the tables, the graphs -

    Speaker 2: To try to check her self. But it doesn't look like she checked herself very well, because she gets different answers every time.

    Speaker 3: Do you multiply the entire problem by 3 when you multiply by 3?

    Speaker 4: Should have [?] instead of divide by 5.

    Speaker 5: And you should have put parentheses around it.

    Speaker 3: X equals Y over 3. When you substitute that in -

    Speaker 5: Oh, yeah.

    Susie (voiceover): Today's lesson was our formative assessment lesson. Solving linear equations in two variables. The students looked at other students' work.

    Speaker 1: She created her ordered pairs through simple math mistake method of -

    Susie: All right! Ladies and gents, we need to work on our post assessment.

    Susie (voiceover): And then they took their post assessment after I gave them their pre-assessment back.

    Susie: Notice the red numbers, please, on your papers. Those are the questions you personally should focus on.

    Susie (voicover): Those five things that they could choose from were the five misconceptions, and therefore feedback questions, that the eighth grade teachers came together the other day after school and wrote together.

    Susie: Okay, you have the rest of the class period to finish. So, on your paper.

    Susie (voiceover): I didn't see as much confusion as I did when they were taking the pre-assessment before the lesson. A lot of them came to the conclusion of, there's only one solution at the end, after our good discussion. I really think that they learned a lot throughout the lesson, and they're going to show growth on their post assessment.

    Susie: Folks, I'm going to need you to wrap that up. If you have already turned in your pre- and post assessment, be sure to turn your calculators in on your way out, and you may go to your lockers.

    Susie: How did your kids react to this, did they do well, do you think?

    Speaker 6: I thought in the classroom they did really well. When we talked about the student samples, they were able to pick up on them right away. They knew what methods they used to solve them, they were able to find the faults with them, so I thought that part went really well. I'm interested to see how the post assessment goes.

    Susie: So, how about we take some time to go through our post assessments, compare them to the pre-assessments, and, if you guys have got your analysis worksheet, and we can find out if our kids actually grew or not.

    Speaker 7: I'm seeing more students like this one here. You can see he says here at the bottom, they sold 3 notebooks and 12 pens, that's just his answer. And you can see at the bottom of this one, he has got work.

    Speaker 6: I just took a glance at the fourth one, and I think there was probably only maybe two in this class who had the right solution last time, and now I'd say probably about 75% of them have the right solution. I haven't looked into the details of if they really showed the work and what they did to find it, but -

    Speaker 6 (voiceover): The lesson and the student work really gives us an anchor to think about future instruction.

    Susie: Is there anything you can think of that maybe we need to hit harder before hand?

    Speaker 8: I think the placement of it made sense, because I think a lot of our kids, with the algebraic portion of it, we've been doing it a lot longer. So, our kids, I don't think there were any kids who did not realize it was a system.

    Susie: That's exactly what I was going to say. My class needed to have more help, more practice on word problems and solving systems before I got into that.

    Speaker 7: Yeah.

    Speaker 6: I think addressing systems -

    Susie: There were some kids who, as we were going through it, were like "Oh! I could have done that? Ohh! I was supposed to do that! I've got it now."

    Susie: The first thing I need you to do is graph both of those linear equations on the same axis, and you have some in front of you. Looks like I've got a lot of folks finishing up the second one. All righty! Here's what I need you to do. Add them together. Could you please do that on your paper somewhere, so I can see it? Grace, can you tell me what you wrote?

    Grace: Negative 3X plus 7Y equals negative 1.

    Susie: Consistency? Is that what everybody wrote? Okay. Now that you've added the equations to create the third equation, I want you to make a prediction. What do you anticipate might happen if we graph that third equation you just wrote on the same axis you put the first two? So now you're going to test your prediction. I need you to graph that third equation on the same axis. Cici, tell them what number you're having a challenge with.

    Cici: Negative 1/7.

    Susie: Did anybody else have to deal with a negative 1/7?

    Class: Yes.

    Susie: Where do we think that is? Negative 1/3, looks good. Rise 1 1/4. [crosstalk] Is your slope - Oh, light bulb just clicked on!

    Speaker 9: To do with all the fractions, I went up 7 and over 3.

    Susie: But if you get rid of the denominator of 7, then you've got 7Y. I thought we could only graph from an equation that was solved for Y, not 7Y. Just Y. So they intersect there, and then - you know what I think just happened? This point somehow made it down there. Do you see how your 1/7 is just barely below zero? So if I go, and I change this down 3, 1, 2, 3, see how it should be right about there? And then 1, 2, 3, 4, 5, 6, 7. See how your point's just a tiny bit off? I think that might have messed things up. Try to get your ruler right on those three points, and see what happens.

    Speaker 10: I've got Y equals X, and that would be up one, over one -

    Susie: Okay. If you got Y equals X, where did the 2 go?

    Speaker 10: The 2 was - oh, wait, that would be 1. Plus 1. That makes it even more confusing. Wait, because it would be divide by - it would be negative 1. I just confused myself.

    Susie: Somebody who has completed the third graph. Can you tell me what your prediction was? Shae, what did you say?

    Shae: That it would also intersect at the same point.

    Susie: The third line would intersect with the first two at the same location?

    Shae: Yes.

    Susie: Did that happen?

    Shae: No.

    Speaker 11: Yes.

    Susie: No, yours didn't, but it did on yours?

    Speaker 12: It did on mine.

    Susie: It did on yours, too? It did, didn't - what happened on yours?

    Speaker 12: It intersected where the other two intersected.

    Susie: Is that what you predicted?

    Speaker 12: Yes.

    Susie: What did you predict?

    Speaker 13: Mine, I said that they would intersect, but at different points.

    Susie: You thought they would intersect at different points? What did you guess?

    Speaker 14: I said that one's going to be perpendicular to another.

    Susie: I was saying that quite a few of you were finding that your prediction the last time was coming true again. That all three equations intersected at the same point. My challenge for you this weekend, is to try to figure out why this is happening.

    Susie (voiceover): Today's lesson was a great experience for the kids. I think the kids enjoyed it, too, and they learned a lot from it.


Susie Morehead
Jenny Barrett


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