Math.Practice.MP4

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP4:  Model with mathematics.

    Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Download Common Core State Standards (PDF 1.2 MB)

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Math.Practice.MP6

Common core State Standards

  • Math:  Math
  • Practice:  Mathematical Practice Standards
  • MP6:  Attend to precision.

    Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Download Common Core State Standards (PDF 1.2 MB)

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Math.7.EE.A.1

Common core State Standards

  • Math:  Math
  • 7:  Grade 7
  • EE:  Expressions & Equations
  • A:  Use properties of operations to generate equivalent expressions
  • 1: 
    Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Download Common Core State Standards (PDF 1.2 MB)

Factor, Expand, and Combine Like Terms
Lesson Objective: Apply properties of operations as strategies to help simplify expressions
Grade 7 / Math / Expressions
Math.Practice.MP4 | Math.Practice.MP6 | Math.7.EE.A.1

Thought starters

  1. How does this lesson help students develop their mathematical vocabulary?
  2. What are the benefits of having students use manipulatives like the work mat and cards?
  3. How could you imagine using this lesson in your own classroom?
7 Comments
I like the way the teacher allows for discovery by demonstrating how she categorized the like terms without telling the students why she arranged her terms as she did. Allowing the students to brainstorm in small groups and progress through the lesson held their attention and engaged all students. If you listen, you will hear a student's comment at the very end saying; "that was fun", best phase ever!
Recommended (1)
Powerful example of students engaged in the Mathematical Practices; I'd also include SMP 3, Construct viable arguments and critique the reasoning of others, which they do when creating the groups. I'd like to suggest that we all are careful about our use of the term "simplify." As I interpret the CCSS-Math, it makes an effort to use "generate equivalent expressions" instead as one form may be more appropriate than another depending on the situation. It's not always desirable to apply the distributive property first, for example; and sometimes, it's easier to connect the expression or equation to the situation it models if it hasn't been adjusted for the traditional simplification. That vocabulary, however, is not the focus of the lesson. The lesson which asks the students to create and justify their mathematical choices is a great example of how engaging students in doing the math and collaborating adds fun, refines their thinking, and will surely raise their retention of the concepts.
Recommended (1)
Great lesson ideas AND provision of cards etc is soooo helpful! Did this with my low class with great success and followed up with a PowerPoint as per explicit teaching practice = SUCCESS!
Recommended (0)
This is a great lesson and very helpful how to design student centered lessons.
Recommended (0)
I am always inspired by teachers who infuse activities into their lessons. Students discussing hw to solve a problem within their groups - "math talk" - is smething we all aspire to. Great job!
Recommended (0)

Transcripts

  • Factor, Expand, and Combine Like Terms Transcript
    Michelle: 02:24: So, are we going to be solving anything today?
    Class: No.
    Michelle: 02:28: We

    Factor, Expand, and Combine Like Terms Transcript
    Michelle: 02:24: So, are we going to be solving anything today?
    Class: No.
    Michelle: 02:28: We will not be solving today because there's no equal sign, so we're just going to be simplifying, making things as small as we can.
    Michelle: Some of you have some cards on your desk, and I have a few here.
    Michelle: 02:26: My name is Michelle Goldberg and I teach math here at Sudbrook Magnet Middle School. This is a class of twenty-seven students. Today was a EQuIP Exemplar Lesson called, "Factor Expand and Combine Like Terms." In the warm-up we had cards and they had different kinds of terms on them. This group today has worked before with variables, they've worked with combining like terms a little bit before but they haven't worked with the XY together and they haven't worked with the X squared or the Y squared.
    Michelle: 03:15: In the original lesson on the website the X and the Y were the only things included. So, in order to ramp it up a little bit for my students I added in the exponents so that they would have some experience with that.
    Michelle: The students were given two columns, a yes column and a no column.
    Michelle: And I'm just going to put mine up.
    Michelle: 03:35: The students watched me put a few of the cards up into the different categories so that they could analyze what I was doing, figure out why was I putting them where I was putting them.
    Michelle: Okay, so talk for about fifteen seconds in your group and see if you can figure out why I did what I did.
    Michelle: 03:50: They had some great conversations at their tables trying to figure out why I did what I did.
    Student: 03:56: Look, since this is a fraction right here and the fractions are all in the yes column and it doesn't have a power. Look it has...
    Student: 04:02: I think she's referring to if it has a variable or not ...
    Student: Right, just like I was saying ... it has a variable right here, and that's a power
    Student: ... the variable's right here.
    Student: Right. That's what I'm saying ...
    Student: They have expressions on the yes side, and on the no side they just have regular numbers.
    Michelle: 04:16: So let's wrap it up: Raise your hand if you have an idea about how I categorized these terms that are up here.
    Student: The yes side they have variables, but the no side they don't?
    Michelle: Good, okay. On the yes side there are variable and on the no side there are no variables.
    Michelle: 04:32: So that helped them to see that some of the terms were just constant numbers and some of them were variable terms, and that there was a difference between the two.
    Michelle: These are considered variable terms because they have a variable, and what are these called?
    Student: Non-variable terms?
    Michelle: 04:48: Okay, good try. Non-variable she says. These are called constant, okay, because there is nothing there that's going to make them change. There's no variable that we're multiplying by that would ever make the variable change. So we have constant terms and we have variable terms. What does coefficient mean?
    Student: Coefficients are numbers that you can multiply by variables.
    Michelle: 05:08: Good. Coefficients are the numbers that we multiply by variables. They come in front of our variables.
    Michelle: They also got a lot of good vocabulary out of that discussion. So we talked about constant, we talked about variable, we talked about coefficients and rational numbers and all of that came out right at the beginning which allowed us to then move forward with using those terms. And I think it was a nice lead in to the next activity.
    Michelle: 05:31: What I want you to get out right now are the little pink ... cards, or papers that have terms on them.
    Michelle There were several different kinds of terms. The first thing they had to do with those was to separate them and put them into categories based on some characteristic that they noticed.
    Student: 05:48: Try to just get it in order.
    Student: This right here is Y variables. This right here is X variables. And this one right here are coefficients.
    Student: These are all like terms. This is like terms. These are like terms. Those are like terms. And then these are the ...
    Student: 06:02: Well technically ... these can't be like terms since this is squaring Y and this isn't. So they're not all like terms.
    Student: These would be like terms.
    Student: Yeah, so these would be like terms.
    Michelle: 06:14: Some of them came up with three groups, and some of them came up with five groups and some of them came up with six groups, and they all had different ways which made sense to group them together.
    Student: Well, we had the rational coefficient, and then we had the variable ... squared. And then we had just one variable group and the two variable group.
    Michelle: 06:33: Okay, so they have constants and then they have the XY's together. They have everything with an exponent together in one group - good - and everything with just and X or just a Y together in one group.
    Michelle: When they were sorting their terms I was hoping that they were going to get to the point where they saw that there were six different categories there. Even though there were a lot of ways they could be put into categories, the ultimate goal was that they would see the like terms there.
    Michelle 07:02: So these are all one category. These are our constants. What's another category that you all have? Terrence?
    Student: Exponents for X?
    Michelle: Okay, so give me an example.
    Student: One six X squared?
    Michelle: 07:16: So by the end they were able to see that X and X squared are not the same thing.
    Michelle: Can the rational coefficient in the front be different?
    Class: Yes.
    Michelle: Yes! What has to be the same? What has to be the same ... Vanessa?
    Student: The variable?
    Michelle: 07:29: The variables have to be exactly the same. So Y and Y squared are not the same thing. You want to be careful of that. Okay, so inside your envelope ...
    Michelle: Once they had that down they worked on a work-mat to create expressions that they were then able to simplify by combining their like terms.
    Michelle: Take your face-down terms. Choose four. Place them on the spaces on your work-mat so that you have an expression lined up in front of you. Okay, so pick four expressions just randomly and put them onto your work-mat.
    Michelle: 08:01: So they would put them on their work-mat. They would come up with their expression and then write the expression down on their workspace and that's where they were able to then identify the like terms and simplify and show their work.
    Student: 08:13: One half equals point five. So negative point eight Y.
    Student: So where would that be?
    Student: Plus, negative ... negative ...
    Student: 08:23: Now what we have leftover is negative three, and one over six ... X to the second power. So we got to take them away since we can't do it. So, this is the simplest thing we can do. So that's it.
    Student: Yeah.
    Student: Let's get some ... Let's get four more. All right. It's already simplified because we can't do anything with it.
    Student: Yeah, it's already simplified.
    Student: Just, put simplified down.
    Student: Okay.
    Michelle: 08:48: I wanted to give them a little bit of practice with the distributive property. That's something that they've seen before but not something that they've seen in this context before. I think it was good for them to be able to manipulate. They liked that they could move it around that way.
    Michelle: After they finished they did a two question skills check, and that was just so that I could see where they were at that point in the lesson. I always walk around while they're working and have little conversations with the kids so that helps me to see how they're thinking.
    Michelle: 09:15: What's the variable on this one?
    Student: X
    Michelle: Squared?
    Student: Mm-hmm (affirmative).
    Michelle: What's this one?
    Student: Y.
    Michelle: Y, what's this one?
    Student: X.
    Michelle: And what's this one?
    Student: Y?
    Michelle: So did you just say the same thing twice for any of those?
    Student: Yes.
    Michelle: Which ones?
    Student: Y.
    Michelle: 09:28: I like to talk to them to say "how did you come up with that?" Because sometimes they're actually on the right track and that they just need one little change in what they did to help them get the right answer.
    Michelle: Three and two thirds. Yes, yes, yes. Yep.
    Michelle: For the most part they were all right on track. They were able to simplify just as I would have hoped that they would be at that point.
    Michelle: After looking at the original EQuIP lesson I decided to make a few revisions to it to suit the needs of my students and my class. We definitely wanted to include some real world problems today so that was one of the things that was not included in the original lesson.
    Michelle: 10:00: I want you to see how we can use this simplifying of expressions to answer some real world questions.
    Michelle: So I modeled a real world problem for them where they were having to label three different things happening in the problem.
    Michelle: I want to take what I know and figure out how I can write it down in a way that's going to help me come up with an expression ...
    Michelle: 10:21: So the problem was about selling chocolate bars for a fundraiser. There were three girls involved. We had to see whether there was some way we could label what they had sold.
    Michelle: So I know something about Leslie, and I know something about Amber and now I have Lily. Do I know anything about what Lily sold?
    Class: No.
    Michelle: 10:38: When I don't know something at all ... what can I do. What can I do?
    Student: Use a variable?
    Michelle: Use a variable, okay. So we can call ... what Lily has sold ... X.
    Michelle: 10:49: I think going forward, they are going to be ready to solve those equations when they need to. To know to simplify it before I move on.
    Michelle 10:58: All right, one thing you learned new today? One thing you learned new? Yes. Jasmine.
    Student: Instead of just having all the X's or Y's you can add them together to make it simplifier. And then you can keep simplifying it to get your answer.
    Michelle: Keep simplifying until what happens?
    Student: Until you can't simplify it anymore.
    Michelle: And why can't you simplify anymore?
    Student: Because there's no more like terms.
    Michelle: No more like terms, very good.
    Michelle: 11:21: I set a goal at the beginning of this year to have as many student centered activities as I possibly could in my lesson. Even though I didn't use the lesson exactly as it was written I was able to take pretty much what was there and just add a couple little things in and use it in my classroom. So it gave me the hands on activities, it gave me the collaboration between the students and it was all right there for me so that was great.
    Student: This was fun!

School Details

Sudbrook Magnet Middle School
4300 Bedford Rd
Baltimore MD 21208
Population: 1005

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Teachers

Michelle Goldberg

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