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.
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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.
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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.
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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)

Applying Understanding of Ratios to Fractions
Lesson Objective: Solve a real-world problem involving ratios and fractions
Grades 6-8 / Math / Modeling
Math.6.RP.A.2 | Math.6.RP.A.3 | Math.7.RP.A.2

Thought starters

1. How does Ms. Morey build off of her students' previous understanding?
2. Why does Ms. Morey give her students independent think time?
3. How does Ms. Morey create opportunities for students to share and learn from each other?
Ms. Morey, Do you happen to have a video for the next class period when you come to consensus? I absolutely love this video as a 7th grade teacher. What were the demographics of your students? Did you have any ELLs?
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I have never found learning to take place from a position of security or great confidence - so well done on choosing a difficult problem to begin with. There is an interesting balance between a strategy that builds confidence and one that motivates learning. Deeper learning I find occurs with the later more than the former in an environment that is motivated by a desire to move from not knowing(insecurity) to knowing(security)
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I want to know if they actually painted the room purple!
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very nice way of teaching
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Wow Jess!!! What a compliment! Your comments made my day. I am so thankful you are finding benefits to you as I share my journey!
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Transcripts

• Applying Understanding of Ratios to Fractions Transcript
Lower Third:
Enumclaw Middle School
Enumclaw, Washington

+++ 00:00:05 +++
Austin: Trying to figure

Applying Understanding of Ratios to Fractions Transcript
Lower Third:
Enumclaw Middle School
Enumclaw, Washington

+++ 00:00:05 +++
Austin: Trying to figure out how many one-and-two-thirds altogether would be in 20 cups.
Girl 1: And the common denominator is six, and I've already timed each side to get six, I'd have to do it to the top.
Crystal Morey: We create a climate where students know that they can share their thinking and that their thinking is validated and that they don't look to me anymore for is their answer is right or wrong. But they look inside themselves.
Card:
Illustrative Mathematics:
Applying Understanding
of Rations to Fractions

+++ 00:00:35 +++
Crystal Morey: Today we get the opportunity of trying to create a perfect color of purple paint. Okay? In fact, I'd like to paint our room. So purple is a better color, and look, just for the task, I wore purple today.
Lower Third:
Crystal Morey
7th & 8th Grade Math Teacher
Enumclaw Middle School, Enumclaw, WA

+++ 00:00:48 +++
Crystal Morey: Our seventh graders have had about three weeks of ratio and proportion knowledge up until this point, have looked at a few of the different models or strategies to show their-- what their thinking is and how they make sense of ratios and proportions. And at the same time, they have not yet looked at anything with fractions.
Card:
Perfect Purple Paint
Perfect Purple Paint is made by mixing 1/3 cup blue paint to 1/2 cup red paint.
How much of each color is needed to make 20 total cups of Perfect Purple Paint?
Crystal Morey: And it says perfect purple paint is made by mixing one-third cup blue paint to one-half cup red paint.

+++ 00:01:11 +++
Crystal Morey: Whenever we introduce a question around ratios and proportions, I try to start by modeling what that actually means in a real-world sense. So many of my students need to actually put it into context of what that measurement tool is used. What does a cup actually look like?

+++ 00:01:27 +++
Crystal Morey: One-half cup red. And together it's going to make our little magenta-y form of purple paint, all right? In order to paint our classroom purple with this purple perfect paint, we need to make 20 cups. And one-third cup and one-half cup doesn't make 20 cups.
Card:
Perfect Purple Paint
Perfect Purple Paint is made by mixing 1/3 cup blue paint to 1/2 cup red paint.
How much of each color is needed to make 20 total cups of Perfect Purple Paint?
Crystal Morey: So our job is in just a second is for you, you're going to go back to your table groups, and you're going to decide how much of each color, "How much of blue and how much of red is needed to make 20 cups of paint?"

+++ 00:01:59 +++
Crystal Morey: When students are introduced with a brand new task, I have noticed that if I don't give them opportunity to think about the task independently first, they aren't able to process it as a group. And so by having students just giving them a quick time to process that, some students might sit there and look at the question for five to six minutes. But they're still processing. So I'm not necessarily in that time period of looking for how much are they getting down on paper. But just providing them an opportunity to
+++ 00:02:27 +++
process so that when they come to the time to talk as a team, they have had an opportunity to think about themselves first.

+++ 00:02:35 +++
Crystal Morey: All right, so you've had some time to individually think about this task. Now's your time to come together as a team, and to create one document that shows your team's best thinking at the moment. You're going to practice coming to Team Consensus. So Team Consensus means a couple of things. And you'll notice those up front. When you come to Team Consensus it means that everyone share their thoughts and ideas. That means you'll have an opportunity to share, and you'll also have an opportunity to listen to other people's thinking.

+++ 00:03:02 +++
Crystal Morey: If we're really going to collaborate on a task together, we've got to really try to convince one another of our thinking. And really the only way to do that is to come to consensus. But students have to not only be good leaders in that they share their thinking, more importantly students have to be really good listeners during that time, and really see it as an opportunity to learn from their peers and to really ask deep questions themselves.

+++ 00:03:26 +++
Boy 2: And we could add one-half to this, and then add another one-third to it. And then this would be two-thirds, and this would be like around, this would be one whole.
Girl 1: We could try this one and see how it works.
Girl 2: I found the common denominator, so I did and one-third out of twelve is four, and then one-half of twelve is six. So.

+++ 00:03:55 +++
Crystal Morey: Okay. How did you know that two went into each box?
Girl 2: I added them all together, and there was ten. And then I divided twenty by ten.
Girl 3: And I got the total as five-sixths, but then I wasn't sure what to use as like for twenty, I wasn't sure whether to use twenty over twenty, or just twenty. So I kind of got stuck there.

+++ 00:04:19 +++
Crystal Morey: I'm going around and I'm documenting which students are doing which different strategies. I'm especially looking for them as conceptions. As a teacher, I believe that the misconceptions tell me more about students thinking that the right answers. So I'm actually honing in most often on kids that have incorrect logic to me, but I need to know what they're thinking and why they're thinking it, so I can improve my practice as a teacher and respond to them better, and not only in that particular moment, but as we come back to whole class and in future lessons.

+++ 00:04:50 +++
Austin: I think I just made a breakthrough. So if there's five whole cups, there's five whole cups times two-thirds by three, you would have six-thirds. Six-thirds would be five-- out of five cups, six-thirds would be blue paint.
Crystal Morey: How did you guys come to consensus on this tape diagram as a model?
Girl 4: Well, so there's twenty total cups. And then we-- I used the-- and then I did three and two. And I knew that three and two was five. And then I divided twenty by five, and I got four.

+++ 00:05:23 +++
Crystal Morey: Where did the three and two come from?
Girl 4: There's one-third and one-half. So I took the denominators.
Crystal Morey: The main misconception that I saw was that students, they would create a model that was three to two, where three was represented by the blue. And two was represented for the red. The problem in that logic is that one-half, which is representing the red amount is a larger fraction than one-third.

+++ 00:05:45 +++
Crystal Morey: Which one's red, and which one's blue?
Jillian: Pink is red, and blue is blue.
Crystal Morey: Okay, but blue originally was one-third. And red was one-half.
Jillian: So, but-- yes, but then we found a common denominator, wait--
Boy 2: Which is--
Jillian: Blue. Part of the one, part of the white.
Crystal Morey: Okay, I want to look at Jillian's work. So Jillian, you're saying this is red to blue, why?

+++ 00:06:02 +++
Jillian: Because so we found a different denominator which was six, because we times-ed it and we found a common denominator. So this is one-half of six, which is three out of six. And this is one-third of six to six.
Crystal Morey: And you said if I put something in each bucket here, I could get to twenty. If all of this represents twenty, what would you put in each bucket?

+++ 00:06:24 +++
Boy 2: Twenty, d-- then you would have to divide it by five, so it would be four.
Crystal Morey: So at the end of the class, we have an opportunity where we come back together. But I don't ask only the proper answers to share during that time. In fact, I think it's important that we have a variety of people share.
Crystal Morey: As we wrap up today, who has a model? Can you share with us Kyler, what part-- actually can I hold it for a second? So which color represents red, and which color represents blue?

+++ 00:06:48 +++
Kyler: Pink represents red, and blue represents green. Green represents blue.
Crystal Morey: Okay. So this is blue, this is red. Who had a model that looks different? Can you tell me what this represents?
Girl 2: That one's the beginning. I did it by the common denominator, and it's six-twelfths for one-half, and four blue for one-third.
Crystal Morey: Okay, so let me pause for second. So your common denominator was out of twelve?
Girl 2: Yeah.
Crystal Morey: Okay, continue on.
Girl 2: And then this, we times-ed it by two.

+++ 00:07:21 +++
Crystal Morey: So here is your-- when you multiplied it, what you got. Okay? We're going to talk about how we might have gotten these when we return next time. And we're going to be sharing our group's thinking. Right now, we have some of us that are in a place where we are convinced of the solution, and some of us are still at a place of confusion. Either way, our job is to come to consensus next class period.

+++ 00:07:42 +++
Crystal Morey: I feel that this lesson was a good start, because I have to believe that we have to start with the messy problem up front. If I start with the messy problem, and I leave them at a point of confusion, we then go back and we really look at the foundational elements of ratios and fractions together, and we revisit this problem in a few weeks, they will know how much they have grown, and how much new learning they have. And through that, they'll really understand the concept of ratios and fractions.

+++ 00:08:07 +++
Crystal Morey: If you're at a point where you don't want to stop--
Austin: I don't want to stop!
Crystal Morey: Austin! That's awesome! I love it! Good job!
#### End of C0804_001002_ENUM_7th_GRADE_FINAL.mp4 ####

School Details

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

Data Provided By:

Teachers

Crystal Morey
Jana Dean
Wendy L. Hughes

Teaching Practice

Three Ways to Encourage Student Collaboration

All Grades / All Subjects / Collaboration

Teaching Practice

Five Ways to Start Your Lessons

All Grades / All Subjects / Planning

Teaching Practice

Three Ways to Practice Goal Setting with Your Students

All Grades / All Subjects / Engagement

Lesson Idea

New Teacher Coaching Cycle: Looking Closely at Text

Grades 9-12 / ELA / Tch DIY

TCHERS' VOICE

Differentiation

TCHERS' VOICE

Class Culture

TCHERS' VOICE

Lesson Planning

TCHERS' VOICE

Move Over Debate, It’s Time to Deliberate

Educating for Democracy