# Series Formative Assessment Practices to Support Student Learning: Formative Assessment: Proportional Relationships

Math.Practice.MP1

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.

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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.

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Math.7.RP.A.3

Common core State Standards

• Math:  Math
• RP:  Ratios & Proportional Relationships
• A:  Analyze proportional relationships and use them to solve real-world and mathematical problems
• 3:
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

## Formative Assessment: Proportional Relationships

Lesson Objective: Formatively assess understanding of proportional relationships
Grade 7 / Math / Proportions
14 MIN
Math.Practice.MP1 | Math.Practice.MP3 | Math.7.RP.A.3

## Discussion and Supporting Materials

### Thought starters

1. How does Mr. Elsdon clarify the intended learning for the lesson?
2. How does Mr. Elsdon use questioning to guide his students?
3. How do the students assess their own understanding?

Mr. Elsdon uses questions to guide his students because he wants to understand the evidence of his students to clarify and elicit evidence of their learning.  His questions are open-ended and require more than a simple answer.  He is careful to guide his students with questions because he is interested in what they are coming up with on their own.  He walks around taking notes to remind himself where his students are mastering the concepts and where he needs to strengthen his lesson.

Recommended (0)
1. How does Mr. Elsdon clarify the intended learning for the lesson? With the whole class Mr. Elsdon reviews the learning goals. The day before he stated he also reviewed the rubric by which they will be graded with.

2. How does Mr. Elsdon use questioning to guide his students? While eliciting evidence of their learning Mr. Elsdon prods just enough; never wanting  to give the next full step. Questions such as where did you get your answer, tell me what you think.

3. How do the students assess their own understanding? Students were able to create/modify the rubric given to them. This allowed students to have self-accountability. Students also had to provide evidence by demonstrating/stating where they were able to find the answers to their questions. Student use their rubric to score one another. I believe the gallery work allowed students to get a visual representation of evidence.

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Mr. Edison clarified the intended lesson by allowing the students to read and understand the learning target prior to starting the lesson and it was visible on their paper at all times. Students were given a rubric and were asked  to self-assess and peer assess within their groups. Mr. Edison visited each group at the posters and at their desk to make sure they were on the right track. The students seemed to accept their peers constructive criticism which made learning easier.

Mr. Edison used questioning as he walked around from group to group. This allowed them to use their knowledge to answer his questions on different levels.

The students were able to assess their own understanding through peer assessment and questions ask by the teacher.  However, the peer assessment was very beneficial.

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Mr. Edison had the students read their learning targets which claried the purpose of the lesson. He guided the students with questioning designed to get them thinking on varied levels. He used multiple techniques and rubrics that led the students to assess themselves and their peers. It was a very effect method of teaching and assessing.

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I love using these type of rubrics. They are great to record what the students are doing and saying. The way the students express themselves and explain their thinking was very impressive. It helps them learn how speak in complete sentences which inevitably will lead the students to write in complete sentences.
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### Transcripts

• Formative Assessment: Proportional Relationships Transcript

Charles: Take a deep breath.
Lower Third
Charles Elsdon
7th Grade Teacher, CREC, New Britain, CT

Formative Assessment: Proportional Relationships Transcript

Charles: Take a deep breath.
Lower Third
Charles Elsdon
7th Grade Teacher, CREC, New Britain, CT
Charles: My name's Charles Elsdon and I teach seventh and eighth grade mathematics and I teach at medical professions in Teacher Preparation Academy in New Britain, Connecticut. First thing we're going to start off, guys, I need some volunteers to read our learning targets. We have four student learning targets. Linwood, could you read number one, please?

Linwood: I understand that multi-step ratios and percent increase/decrease problems can be solved using proportional relationships.

Charles: Beautiful, very nice job. When you do a read aloud with the learning goals, it involves the students a little more and so it's just, they're not listening to me the whole time. That way, they're listening to their peers. For the clarifying portion, that ties into what we did yesterday with the rubrics. I clarified my expectations, what the goals were, the success criterion. We talked about all those different things yesterday and they were actually on the rubric, so they can look at it and refer back to it today. So we discussed these yesterday and what they mean in a little more in depth, as well as creating these rubrics for today. So that's what you guys are going to be using later on to score each other's projects.

Then we took the learning targets and I showed them the rubric that I had designed for them, how they would be scored on my end. And what I did was, I gave them a little bit of free rein to change the words a little bit. So like for instance, I had "at or exceeds expectations" on my chart, and they changed it to "the Nobel prize" because they just wanted to be a little more a part of it. So what are some real world examples or some non-examples about proportional relationships. Makiah.
Makiah: One real world example is miles per hour.

Charles: Miles per hour, okay. That's a nice rate, right? Rates are great and they help us with proportional relationships, but what's a proportional relationship? We've talked about this before, guys. I see Rafa doing the right thing. Look at your notes, right? Take a look, quick. Quick look, quick look. What is that?
Student: Like if you had one pound of coffee for \$3.
Charles: Okay, but what would be proportional to that? Grace?
Grace: Two pounds of coffee for \$6.

Charles: Beautiful job, okay. Also during the lesson today, I had brought up the success criteria. So these were our ultimate goals, you guys need to be able to do this. This is our do part, or in other words, it says, I can. Notice how these are a little bit different than what our learning targets were, right? Your learning targets are up top on the top of your rubrics. They're a little bit different. There's a big difference in word. It says, "I can," right? On the rubrics, it says, "I understand." That means you're getting the idea, but "I can" means that you can demonstrate it. You can show me, right? So I can analyze proportional relationships and solve

multi-step percent increase and decrease problems. So today, we were dealing with percent increase/decrease markup and discount. Each group was set up so that they all had different questions that were similar questions, so they were differentiated upon their level. They had to try and solve the questions and develop a poster and then we did a gallery walk at the end where they would go to different groups, they would score each other with the rubrics that we had created before, as well as, I was scoring them as well on my own rubric, and then at the end of the class, we

brought it all back together. So I'm going to let you guys go ahead and do your thing now, which is discuss,
come up with solutions. What you need to do is make a poster and practice speaking, all right, because you are going to share that, like I told you yesterday, with the other groups as you come around.
Card

Charles: There's four of you here. You guys have resources. You have your notes, you have your brains, right? You're all really smart, so what can we do? Now underneath your table carts that say your group's name on them, all right, there are two questions. Your task today is, you're going to begin with trying to find a solution to those questions and I want you to find more than one.
Lower Third
Charles Elsdon
7th Grade Teacher, CREC, New Britain, CT

Charles: So eliciting the evidence is when you're walking around. You're looking for, what evidence are they bringing up? So instead of clarifying the questions that they have, you want to make sure that you see what they're coming up with on their own. What did you divide it by?
Student: One twenty-five divided by .10.
Charles: Where did you get .10 from?
Student: The ten dollars.
Student: Because it's 10 percent.
Charles: The 10 percent, right? So you changed the percent to a decimal and you got 12.50. Does that make sense if you think about it?
Student: No, it doesn't.
Charles: If that would be the original price and there's only a 10 percent discount, okay?
Student: You multiply?

Charles: I don't know. Tell me what you guys think. Keep exploring. I was checking in to see what they were coming up with, what was the information that they were communicating to each other, what were they coming up with on their paper?
Student: Ten.
Charles: What was that question again, before you guys move on?
Student: The sale price is \$80 after a 10 percent discount. What was the original price of the item?
Charles: Okay, so what are we going to be looking for here?
Student: The price before the discount.

Charles: The price before the discount, good. Nice job, ladies. Whenever you're guiding a student, you want to make sure that you are just providing just enough prodding with your questions to make sure that, "All right, well maybe I should think about that a little more," or, "Maybe I can go along that path, but what would be next?" You never want to give them the full next step. You want to just kind of guide them with your questions. Where is your decimal?
Student: Here. Do we have to move it to-- ?
Charles: Ah, well it depends on what you're going to do. What are you going to do with it?
Student: Divide.
Charles: You're going to divide? So would I just use the percent number?

Student: Oh, wait, let me go back a minute.

Charles: And then they needed to create their posters. So on the posters is the other way that they can show the evidence, and therefore display it to the other students and talk to the other groups about the evidence. I split the groups in half, so half the group would stay and present their poster to the visiting group. And so the visiting group would come in, they have two minutes, that's it. Get your conversation in. You have to talk, you have to be quick, but you also have to be able to ask questions.

Then it's two minutes next, you move. I need each group to decide which two people will stay at their poster to explain what they did and which two people will travel. Those who are traveling, you need to make sure you bring your rubrics, because we discussed how you're going to score the other tables.
Student: Because the sale price is 20 percent, plus then there was no price, so we made a two-step equation. And we know that if you want to find a percent of a number, then you multiply the number by the percent of the decimal.

Charles: Remember we made up the scores, so needs improvement, emerging, proficient and the Nobel prize. Okay, those are our different levels. Since the students were involved with creating the rubric, it creates that more sense of self-accountability. So they are not only scoring the other groups as they go along on the rubrics, they're scoring their own groups. So therefore, they're really involved with the whole process and it just brings it back on themselves instead of me just scoring them with the one grade. "All right, you did a good job today. Check." It's a lot more involved.

Student: We have to multiply 140 times...
Student: Right.
Student: Uh.. wait.
Charles: I saw the light bulb. Talk to your group mates about it. Let's see what happens.
Card
#### End of C0704_005030_Charles_Elicit_Final_SD.mp4 ####
Student: 72 is a lot far from 800, so I think we should keep going.
Charles Elsdon: From interpreting the evidence I did a lot of listening. I just wanted to see what those groups were saying in the questions, the responses that the other groups were giving back to them.
Student: You didn’t fully understand like the present increase/decrease did you?
Student: I did understand, did you understand it?

Faizah: You explain what you did on your poster so the people can understand what you did, so they know how to do it too, and then they grade you on how well you understood it, and how well you could do the problem.
Charles Elsdon: When your peer’s evaluating you, for instance if I had another teacher come in and give me some feedback, that helps me make myself better. So it gives me a little more intrinsically motivated to just make myself better, and make sure I really understand what I’m doing. And that goes the same thing for the kids.

Charles Elsdon: Those who are traveling you need to make sure you bring your rubrics because we discussed how you’re going to score the other tables.
Linwood: The rubric we worked on was like the scores that we would give. Like if it was as one it was like you get it, but you’re like not there. And three is like you fully understand it.
Charles Elsdon: Linwood, you have some great questions, very nice questions. So as you’re looking up here, right, do you think that there could be another way to find the answer?
Student: Yeah.
Charles Elsdon: Or do you think just one way is it?

Student: No there could be more ways.
Student: There could be much more ways.
Charles Elsdon: There could be multiple ways, right? I like that.
Student: We only figured out one.
Charles Elsdon: And you only found one, but that’s okay, we have a limited time in here, right?

Charles Elsdon: I was scoring them as well on my own rubric. I was looking for very key things. So all these mini formative assessments that you do you’re listening for key words, you’re listening to their idea patterns, you’re listening to how they’re discussing to the other group members in their group, and how they’re critiquing the other groups as they listen.

Charles Elsdon: So what I’d like you to is go back to your seats please, so we can bring this back together, and we’ll wrap up.
Charles Elsdon: What did you notice that most groups did? What did they use?
Student: They used a two-step equation.
Charles Elsdon: Yeah they used the T-chart, right? Well when we go back to our success criteria’s what do I need to make sure that we can do? Grace?
Grace: A proportional relationship.

Charles Elsdon: A proportional relationship, good. So we were just a little off when it came to that one for most of the groups, we stuck with equations because we’re a little more comfortable with equations, right?
Charles Elsdon: Well I was a little surprised that everybody kind of came up with the same solution path, which that could’ve been something on me because I may have been pushing in one direction too much, and that’s something that I’ll have to think about tonight when I look at their evidence.
Faizah: We didn’t get the proportional relationships part, but we did do the increase and decrease.

Charles Elsdon: So what I need you guys to do is leave your rubrics and your scrap paper that you used on your tables please. I need those, all right, ‘cause I’m gonna look at those tonight, and make sure what we do tomorrow is appropriate.

Charles Elsdon

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