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ratio of devices to students in each class K-4 beginning in 2018

What is a number talk?

Article: Number Talks Overview & Teacher Role

This summary sheet outlines what a number talk is and gives an overview of the routine along with a description of the teacher's role during a number talk.

Video Ruth Parker & Cathy Humphries--Making Number Talks Matter

Meet the authors of Making Number Talks Matter as they discuss the power of number talks.

The following protocols can be used with any number talk video.

This protocol helps teachers identify the presence or absence of the number talk routines outlined in Chapter 2 of Making Number Talks Matter by Cathy Humphreys and Ruth Parker.

Connecting to the number talk planner in Appendix A of Making Number Talks Matter by Cathy Humphreys and Ruth Parker, this protocol encourage teachers to consider the questions and recording strategies modeled in the video.

This chart lists the guiding principles of number talks as defined in Chapter 3 of Making Number Talks Matter by Cathy Humphreys and Ruth Parker to allow viewers to record teacher moves aligned with each principle.

Beyond Fingers: Place Value & the Numbers 11-19 (8 min)

Math Solutions Number Talks K-2 Rekenreks (5 min)

Math Solutions Number Talks 2.2 Addition 16 + 15 (3 min)

Number Talks Mental Math 2nd Grade 006 (5 min)

Third Grade Mental Math (6 min)

3rd Grade Number Talks 2015 (5 min)

3rd Grade Amber Taylor Number Talk Multiplication (8 min)

Mental Math, 3rd Grade in Spring (13 min)

Originally from the Stanford University online course "How to Learn Math for Teachers and Parents," this video features Jo Boaler leading a number talk with college students solving 12 x 15.

Document: Number Talks--Kindergarten
Based on the book Number Talks by Sherry Parrish, this document from shares an overview of number talks before launching into lists of strings for a variety of computational strategies and models appropriate for kindergarten.

Document: Number Talks--First Grade
Based on the book Number Talks by Sherry Parrish, this document from shares an overview of number talks before launching into lists of strings for a variety of computational strategies and models appropriate for first grade.

Document: Number Talks--Second Grade
Based on the book Number Talks by Sherry Parrish, this document from shares an overview of number talks before launching into lists of strings for a variety of computational strategies and models appropriate for second grade.

Based on the book Number Talks by Sherry Parrish, this document from shares an overview of number talks before launching into lists of strings for a variety of computational strategies and models appropriate for grades 3-5.

This site, hosted by middle school teacher Fawn Nguyen, shares number talks used with middle school students, along with examples of student responses and a page with suggestions for giving number talks in the middle school classroom.

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Number Talks - Ellensburg School District 401

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ratio of devices to students in each class K-4 beginning in 2018 Article: Number Talks Overview & Teacher Role This summary sheet outlines what a number talk is and gives an overview of the routine along with a description of the teacher's role during a number talk.

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Number Talks Lesson Plan

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This is the first in Crystal Morey's Getting Better Together series, Making Number Talks Matter Book Study. Crystal and all the Teaching Channel Laureates are going public with their practice and seeking support in getting better from colleagues and the Tch community.
Growing up, I was a struggling reader. I had trouble retaining what I read even though my fluency was impeccable. So it's no surprise that in school, I gravitated towards math. I was the "smart" math student. On timed tests, I would be the first done and would turn my paper over with a flourish as I publicly displayed my quickness.

The next few years were more of the same: memorize algorithms and apply them as quickly and efficiently as I possibly could. I became very accustomed to the process.

I remember, though, shopping in a store with my mother and trying to determine a discounted price. I would literally find a piece of paper in my bag, write down the problem, and solve it. My mother would look at me puzzled, wondering what had taken me so long. She used different mental strategies and could quickly apply math - and she was more accurate and faster than I was.

The fact that I struggled so much on real world applications in math always stayed in the back of my mind. And then I met Ruth Parker.

I remember Ruth leading a group of us in a number talk - a short, daily routine that provides students with meaningful ongoing practice with computation - and asking us not to use our pencils. Was she kidding? I was horribly addicted to pencil and paper. I solved everything with memorized algorithms. I knew it was going to be a challenge. I remember one of the first number talks was related to subtraction. Wanting to use borrowing, I literally imagined the algorithm in my head and tried to do some "air borrowing" with fingers behind Ruth's back. I got three different answers as I attempted to solve the problem.

During the share-out time, I watched as other teachers articulated strategies that actually made sense. Their strategies were efficient, easy to use, and the reasoning behind why they did what they did was clear. I started trying out my classmates' strategies and within a few number talks, I finally felt that I was making sense of the math behind subtraction.

Confronting My Own Tendencies

When I became a teacher, I realized that before I could begin number talks in the classroom with my students, I had to confront my own tendencies, my own insecurities, and my own practices that were leading to memorization over sense-making.

In the three years since my first number talk with Ruth, I've challenged myself on a daily basis to think about numbers differently. You won't find a sticky note in my purse or me "air borrowing." I have efficient strategies and they make sense to me. Moreover, I have seen this happen for my students. Those kids who were previously paralyzed by memorized algorithms now have the tools to solve problems critically. By using clear explanations, students are becoming more precise in their language, which also helps to deepen their understandings. They are constructing viable arguments in order to convince others that their processes makes sense. This instructional method sets the stage for rich tasks that allow for deep thinking and conceptual understanding. And it's in these number talks that I see the Standards for Mathematical Practice come to life.

With only one year of number talk implementation behind me, I am very much a novice. I've begun to see their value for my students. Yet, I have little confidence when a number talk seems to bring about incorrect number sense. I wonder how - and how much - to question an individual student. I wonder how to pick the just-right number talk.

Number Talks Book Study

With the help of my fellow Teaching Channel Laureate, Kristin Gray, I've decided to launch a Making Number Talks Matter Book Study, which will take place on social media and on Teaching Channel Teams, and is based on the book by Ruth Parker and Cathy Humphreys. I hope to receive feedback and insights that improve my daily practice as I post reflections and videos of number talks I've led. I plan on being vulnerable, documenting number talks that didn't go as I had anticipated, and using these "real life" moments to connect and grow with colleagues like yourselves. (I asked my students what they wanted filmed and shared. They said the number talks where it almost feels like a debate, where the answer isn't clear, and they're struggling in sense making.)

Also as part of the book study, I'll be meeting with Ruth herself to analyze a number talk of mine and give me feedback. I want to show that rich feedback is imperative for me as I continue to grow as an educator. I want my confidence in responding to and documenting student thinking to increase, with the overall goal of having a positive impact on my students' learning.

On October 5th, we begin our 11-week journey with over 500 teachers, from all over the world, on the Making Number Talks Matter Book Study. The group ranges from teachers, to instructional coaches, to consultants who live and work in the United States, yes, but also Australia, Singapore, Saudi Arabia, and the United Kingdom. More colleagues than I could have ever imagined are joining us on this journey! (See the map of registrant locations below.)

If you weren't able to register in time, follow along on Twitter (make sure as well to follow me and Kristin) or Facebook, and here on Teaching Channel's blog. You're always free to ask questions or add your thoughts via the comments section below.
Crystal Morey is a middle school math teacher in Enumclaw, WA. She has served on the county's Common Core transition team and is a teacher leader on a statewide project with Washington State's Office of the Superintendent of Public Instruction, and the Mathematics Education Collaborative. Connect with Crystal on Twitter: @TheMathDancer.

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Making Sense of Math Through Number Talks

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This is the first in Crystal Morey's Getting Better Together series, Making Number Talks Matter Book Study . Crystal and all the Teaching Channel Laureates are going public with their practice and seeking support in getting better from colleagues and the Tch community. Growing up, I was a struggling reader.

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Developing Mathematical Practices and Deepening Understanding, Grades 4-10

Cathy Humphreys and Ruth Parker

Mathematics Professional Reading

In my district, Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5 by Sherry Parrish has been a major resource for our primary and junior teachers to promote proficiency and fluency in number sense and numeration for the past five years. Now, Cathy Humphreys and Ruth Parker - who provided words of praise for Parrish in the front matter of her book - have released a book of their own with a focus on grades 4 to 10.

For those keeping track, Parker is one of the creators of Number Talks back in the early 1990s, and Humphries is known for her work extending them to intermediate and senior grades. Now, they have come together to continue the Number Talk love in the junior and intermediate grades on a quest to deepen student understanding and proficiency with number.

Jo Boaler writes in the foreword:

Teachers can watch a video of an expert teacher giving a Number Talk and believe that they are simple to teach, because they can appear deceptively easy. Yet when teachers begin to teach Number Talks themselves, many questions arise: What do they say when students share an incorrect solution, or when there is a mistake in their work? What do teachers do when students have no methods to share? How do teachers know the best problems to use for Number Talks? Where do they find examples of Number Talks for middle and high school students? These questions, and many more, are answered in this book.

In the first chapter, the authors explain what Number Talks are and why they are so important. They highlight some of the many student struggles in math and attribute them to how arithmetic is often taught as a set of rules and procedures to be remembered rather than concepts to be understood. With this, they are also quick to give respect to algorithms for their importance as reliable and efficient tools necessary to learn over time. However, they discourage a rush to the algorithm due to the "compactness" of algorithms that "hides the meaning and complexity of the steps involved" as quoted by the Hyman Bass work Computational Fluency, Algorithms, and Mathematical Proficiency, 2003. They give simple examples such as adding fractions, employing the subtraction algorithm, and even algebraic concepts such as expanding a perfect-square binomial.

The authors move on to presenting the reader with what is "as close to a recipe as you will find in this book" through an 8 step routine that can be used to facilitate an effective number talk in the classroom. With each step includes rationale and some key tips to remember, which are compact enough for teachers to keep in their daily lesson plan. Teachers are encouraged to begin using dot cards as a starting point as this is very non-threatening for teachers and students and can still provide an opportunity to share mathematical thinking through the use of numerical expressions incorporating different operators and their appropriate order. After providing teachers with an exemplar and a transcript to model the specific Number Talk, there are key ideas to promote successful use of Number Talks in the classroom.

The remainder of the book is split up into different strategies for different operators across grades 4 to 10 including addition, subtraction, multiplication, division, fractions, decimals and percent. Within each of these chapters, specific strategies are offered with exemplars and even suggestions for particular Number Talk questions that promote using one strategy over the other.

If I could provide any constructive criticism for this book, it would be around the use of visual (and concrete) manipulatives. While the use of arrays and area models is shown in some cases for multiplication and division for numerical and algebraic expressions and equations, I would argue that it would be very helpful for teachers to have more visual tools and representations at their disposal to assist students with visualizing (or spatializing) their understanding.

Overall, this is a great read, especially for those teachers who falsely believe that there is no place for Number Talks in their junior, intermediate or even senior level math classroom.

Buy the book on Amazon.

Connect with Ruth Parker on Twitter or at the Mathematics Education Collaborative (MEC).

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Making Number Talks Matter - Cathy Humphreys and Ruth Parker - Book Review

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Mathematics Professional Reading In my district, Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5 by Sherry Parrish has been a major resource for our primary and junior teachers to promote proficiency and fluency in number sense and numeration for the past five years.

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I honestly cannot talk about the teaching and learning in my classroom, or across our school, without highlighting Number Talks. I cannot recall the very first time I started Number Talks, but now I cannot imagine my math class without them. My Number Talk journey began with Sherry Parrish's book and had continued to grow through reading Cathy Humphreys' and Ruth Parker's most recent book, Making Number Talks Matter . I implement Number Talks on a regular basis, reflect through writing after and have presented at both NCSM and NCTM around Number Talks. I simply cannot say enough wonderful things about them!

For this reason, it was not surprising when I had the opportunity to meet my fellow Teaching Channel Laureates this summer, that Crystal (@themathdancer) and I struck up an immediate conversation about Number Talks when we began chatting instruction. As an elementary school teacher, I often have middle and high school teachers ask what Number Talks could look like in the secondary classrooms. Would the setting be the same? What would example problems or strings look like? How does the content focus change? How do we get students at that age to engage in these mathematical conversations?...etc. So many questions that I still am trying to wrap my head around.

Fortunately for me, Crystal is a middle school teacher using Number Talks, so our conversations gave me great insight as to how they look and feel in the middle school. We talked extensively about the Number Talk course she had taken with Ruth Parker and shared students' conversations during Number Talks in our classrooms. It was so exciting to see the connections between our students' experiences and the path Number Talks can take after my students leave me and enter the middle school. While I would love the opportunity to plan and observe Crystal's classroom, and her to visit mine, she lives in Washington state while am on the east coast in Delaware, so our opportunity to collaborate face to face is not a reality.

Enter the wonderful world of technology, amazing resources available through The Teaching Channel, incredible teachers around the world wanting to learn and grow every day, and Ruth and Cathy's new book Making Number Talks Matter ! Now, Crystal and I do not have to be on this amazing journey alone, but instead we have the opportunity to create an experience in which other educators everywhere can join us!

Together, we have planned and structured a book study unlike any other! Flexible to fit your needs, full of valuable resources, and completely FREE!

Beginning October 5th, each week will be dedicated to one chapter of the book. Conversations will happen on and , videos will be posted in our Teaching Channel Teams group, and we will even have a guest appearance by Ruth Parker herself! It is guaranteed to be an amazing learning experience that can only get better with your participation!

For more information and to register for the book study, follow the links below:

We look forward to learning with you and Getting Better Together!

-Kristin

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Making Number Talks Matter - Book Study!

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I honestly cannot talk about the teaching and learning in my classroom, or across our school, without highlighting Number Talks. I cannot recall the very first time I started Number Talks, but now I cannot imagine my math class without them. My Number Talk journey began with Sherry Parrish's book and had continued to grow...

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You searched for book monday number talks - Math Coach's Corner

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Three years ago I had the pleasure of hearing Sherry Parrish and Ann Dominick speak at the NCTM Annual Meeting in New Orleans. My campus had been using Number Talks in all of our K-5 classrooms for several years, and I knew first-hand how dramatically they could impact computational fluency and mathematical reasoning.

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Teacher modeling the use of Number Talks to help develop numerical reasoning and strategy development.

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Sherry Parrish: Number Talks: Building Numerical Reasoning

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Post Talk Number Talk
ELYSHA PASSEGGI: So this morning we're going to do a number talk and we're going to talk about the same kind of ideas that we were talking about on Wednesday. So I want to just look back at what we discovered and talked about on Wednesday. Does anybody remember what we kind of realized when we were talking on Wednesday? Erik?

STUDENT: We kind of did like the problems that you had to try to see if it might be true or false without thinking and just looking at the numbers and you knew that they were true or false.

ELYSHA PASSEGGI: So was it without thinking or was it without something different?

STUDENT: Without something different.

ELYSHA PASSEGGI: What was it? Rebecca?

STUDENT: Without a calculation.

ELYSHA PASSEGGI: Without doing calculations. And how many people find that challenging? Did anybody come up with something that made it a little bit easier to not calculate the numbers? Did we talk about anything that made it easier to not calculate? Olivia?

STUDENT: Looking at it through relationships.

ELYSHA PASSEGGI: What do you mean by that?

STUDENT: Um, the numbers are each one less than thirty-two and thirty-three and forty-four and forty-three.

ELYSHA PASSEGGI: Okay, so you noticed a relationship between forty-four and forty-three, that forty-three was one less than forty-four and thirty-two was one less than thirty-three? And that was helpful?

STUDENT: Yeah.

ELYSHA PASSEGGI: Anybody has anything else to add? Christopher?

STUDENT: It's a little bit different from the fact families.

ELYSHA PASSEGGI: Okay and how did that help you?

STUDENT: Well, say that I knew like um...um say that this was just a made up problem and I knew three, four, if you multiply them, they equal twelve then (inaudible). What times four equals twelve? Did they...it's like a fact family, you would just have to answer the question.

ELYSHA PASSEGGI: Did knowing the multiplication of the fact families helped us with these problems? STUDENT: No, kind of.

ELYSHA PASSEGGI: Not as much? Kind of?

STUDENT: Not as much but kind of.

ELYSHA PASSEGGI: Ok. Alright, so I want you to think about that today when we're solving some other ones and see if fact families help you, okay. And we will talk about that again. Okay, I want to show you a couple...we're going to do the same type of thing. I'm going to show a true or false problem and I want you to take some time. Don't put your hands up right away. I want you to think about it and think if the problem is true or false without doing calculations in your head okay. And then we'll talk about how you know. Alright, the first one is 5 + 7 = 12. Just keep your hands down for a second and think about it. Is it true or false and how do you know? Okay, who thinks they know? Annabelle what did you do?

STUDENT: I split the five to three and two because seven plus three is ten and I just added the leftover two.

ELYSHA PASSEGGI: Okay, so you split this into a three and a two and you knew seven plus three equals ten. And then you added the two to get twelve. Did anybody do it in a different way? Erik?

STUDENT: Well I knew that six plus six also equals twelve and then I switched one of the digits...well I took one off the seven and put it onto the five and added them together.

ELYSHA PASSEGGI: Okay, so you traded one...you added one here like that?

STUDENT: Yeah.

ELYSHA PASSEGGI: Okay. Alright, does everyone agree that this is true? Do you disagree?

STUDENT: Okay, let's look at the next one. Now see if you could figure this out without doing any calculations. 21 + 39 = 50. True or false? Daniel what do you think?

STUDENT: I think it's true because nine plus one equals ten and...wait, no.

ELYSHA PASSEGGI: Okay. So what did you just realized that made you change your mind?

STUDENT: Well because at first...because thirty plus twenty would equal fifty but that equals sixty because it takes the nine and one from the thirty-nine and the twenty-one is ten, which would add ten to it.

ELYSHA PASSEGGI: Okay, so first you started by looking at the ones and then I realized that you said "oh wait a second" and I think that what you realized is that when you looked at the twenty and thirty that you already had fifty, but there was going to be more to add to that, which would give you more than fifty. Am I correct? Okay, anybody want to add something different to this one or saw this in a different way? Heather?

STUDENT: Well, I thought it as like you just took the one from the twenty and move it to the thirty-nine, which made you have forty and twenty and that's how you sort of know the answer is not right because it's ten over fifty.

ELYSHA PASSEGGI: Anybody see it in a different way and wants to add anything else? Ellie?

STUDENT: Well I kind of did it differently because as soon as I read it, I was almost positive it was fifty then I did it again and then I got sixty. So you can't go through it really fast because then you don't get all the calculations.

ELYSHA PASSEGGI: Okay, so when you just glanced at it you thought that it was true and then when you took a second glance and you kind of started calculating, you thought maybe it wasn't true, correct? Christopher?

STUDENT: Well I thought this was false and what I did is that I rounded the numbers. So since I added one to the thirty-nine to make it forty and I subtracted one to the twenty-one to make it twenty, that basically is like doing nothing because zero plus one equals one and minus one equals zero. So zero zero is the same number and so well I thought that...

ELYSHA PASSEGGI: These zeros?

STUDENT: No, it's just like a random equation.

ELYSHA PASSEGGI: Okay, so I'm just wondering then you're saying that zero plus zero...I'm just wondering how that is connected to this. I'm just not understanding.

STUDENT: Oh. Well that's not what I mean. Like it's just like if you add one and then subtract one, it's like basically equals nothing.

ELYSHA PASSEGGI: Oh, so you're saying that because you added one here and you subtracted one from here that it kind of was cancelling each other out, or that you were doing the same thing to each number...kind of the opposite?

STUDENT: (inaudible)

ELYSHA PASSEGGI: Okay, that makes sense to me now, thank you. Anybody disagree or want to add anything else? Erick?

STUDENT: Well I knew that twenty plus thirty equals fifty, so then if you even had like a one or a nine that you already will be over. So I knew that that was false.

ELYSHA PASSEGGI: Okay, so looking at this and then knowing the one, which I think Daniel discovered when he went back to it. Okay, one more. Twenty seconds to think about it. 62 + 76 = 238. True or false and why? What do you think Kate?

STUDENT: Um, I think it's false.

ELYSHA PASSEGGI: Why? STUDENT: Because sixty-two and seventy-six will equal something in the one hundreds. Because to get two hundred you need at least one hundred and one hundred to equal two hundred, so I think it's false.

ELYSHA PASSEGGI: Okay, anybody want to add to that or saw it in a different way. Taniya?

STUDENT: Well I knew that six plus seven equals thirteen. So that would equal one hundred and thirty and then...two plus six equals eight so you may need a lot more to make it at least two hundred.

ELYSHA PASSEGGI: Okay, so kind of what Kate was saying, that it looked like it was going to be in the hundreds.

STUDENT: Yeah.

ELYSHA PASSEGGI: Okay, anybody see it differently? Rosie?

STUDENT: At first glance I thought it was true but then I wanted to just like make sure, so I added the sixty and the seventy and I got a hundred and thirty. And then like "what, that couldn't work." And then six plus two equals eight and that's actually one hundred and thirty-eight, not two hundred and thirty-eight.

ELYSHA PASSEGGI: I have a question. Looking at this problem here, when you see this number two hundred and thirty-eight, does this look like a reasonable answer for this problem?

STUDENT: No. The numbers are too small.

ELYSHA PASSEGGI: Ah. Aden what do you mean by that, the numbers are too small?

STUDENT: They have to be bigger to get to a bigger number. Those need like...I don't know. I don't think those are big numbers to get to two hundred thirty-eight. They just don't look big enough.

ELYSHA PASSEGGI: Okay, Frederick do you want to add to that?

STUDENT: Um, yeah. Well, I did two things. Um, I did two different things. So first I already knew that that was a hundred and thirty-eight, but also two two digit numbers no matter what, if it's addition cannot be more than one hundred ninety-eight.

ELYSHA PASSEGGI: So you're saying that no matter what, two two digit numbers...

STUDENT: Added together.

ELYSHA PASSEGGI: Added together can never be more than...

STUDENT: A hundred ninety-eight.

ELYSHA PASSEGGI: Interesting, does anybody agree or disagree with that? What do you think? I think that's an interesting comment. I don't know if it's true, I haven't tried on everything. Why would this be true?

STUDENT: Why wouldn't it?

ELYSHA PASSEGGI: Or why wouldn't it? Maddy?

STUDENT: I think it's true because the highest two digit number you could have is ninety-nine and ninety-nine and ninety-nine is a hundred and ninety-eight.

ELYSHA PASSEGGI: Interesting. So using that rule, if it was true and you saw two two digit numbers and then an answer that was larger than two hundred, could you assume right away that it's false?

STUDENTS: Yes.

ELYSHA PASSEGGI: If Frederick's rule is true?

STUDENT: ...exactly the last number but it's the highest.

ELYSHA PASSEGGI: The highest?

STUDENT: Two digits.

ELYSHA PASSEGGI: Two digits?

STUDENT: The highest two digit.

ELYSHA PASSEGGI: Okay. Alright, I have put a question up here that is the same question that I asked you the other day. Can you figure out in your head -- now without calculating, that means without adding this and taking from here and putting the...try and just look at the relationships of the numbers. Can you figure out in your head if these problems are true or false? Ready? Okay. Take a second to think about it. Don't put your hands up, just think about it for a second. 32 + 28 = 33 + 27. Don't put your hands up yet, wait. Think about it.

STUDENT: I think it's true.

ELYSHA PASSEGGI: Why?

STUDENT: Because if you add...well I just think it's true because the three would fit into the seven on thirty-three plus twenty-seven and then the two would fit into the eight. They both equal ten so I think it's true.

ELYSHA PASSEGGI: So what did you just do to figure that out?

STUDENT: Two plus eight is ten and three plus seven...

ELYSHA PASSEGGI: But what strategy did you use to figure that out?

STUDENT: Well, calculate.

ELYSHA PASSEGGI: Doing calculations. Which you can definitely do to get the answer but I'm just wondering did anybody did it a different way without calculating?

STUDENT: Well the thirty-two and the thirty-three are one number apart.

ELYSHA PASSEGGI: Okay.

STUENT: And then the twenty-eight and the twenty-seven.

ELYSHA PASSEGGI: Are what?

STUDENT: Are one number apart.

ELYSHA PASSEGGI: Okay, if I have from thirty-two to thirty-three, you're saying that the difference is one? So what changed from thirty-two to thirty-three?

STUDENT: It got bigger.

ELYSHA PASSEGGI: It got bigger by one? So could I record that by saying that? Does that make sense? And then what did you notice about the twenty-eight and the twenty-seven?

STUDENT: Same thing.

ELYSHA PASSEGGI: The same thing? Okay, so from twenty-eight to twenty-seven...

STUDENT: It got minus.

ELYSHA PASSEGGI: Oh, minus. Interesting. Kate what do you think?

STUDENT: I think they equal the same thing because if you put the twenty-eight on the thirty-two it would be thirty-three and it would balance both sides.

ELYSHA PASSEGGI: Okay wait, wait, say that again.

STUDENT: If you put one from the twenty-eight onto the thirty-two then the thirty-two would become thirty-three and it would be the same as the other side. And if you put one from the thirty-three onto the twenty-seven then it would become twenty-eight and both the sides would be equal.

ELYSHA PASSEGGI: Okay, so you're saying -- you're looking at the relationship of these two numbers and these two numbers, where Rose was looking at the relationship of these two numbers and these two numbers. Is that correct?

STUDENT: I was kind of looking at those relationships also. The other numbers help.

ELYSHA PASSEGGI: Okay, so you're taking one from this and adding it to that and then you're taking one from here and adding it to that. And then this one you're going this way right? So you're taking one from here and adding it to there, so that both sides would be twenty...thirty-two plus twenty-eight? Okay, Heather?

STUDENT: When Rose did it, the one sort of canceled each other out like on hands on equations, so that they basically just canceled each other out.

ELYSHA PASSEGGI: This way? Okay, so you're saying that because the change is doing the opposite thing, they're canceling one another out?

STUDENT: Yeah.

ELYSHA PASSEGGI: Okay, the same kind of ideas as hands on equations.

STUDENT: Yeah.

ELYSHA PASSEGGI: Okay, so do you then agree that this is true or false?

STUDENTS: True.

ELYSHA PASSEGGI: Okay. Alright, let's look at another one. Think about it. See if you think this is true or false. Think about it. Don't put your hands up yet. Take a second to think about the problem. 41 + 36 = 44 + 32. Does anybody see anything here that would help us figure this out? Okay, so you were rounding. So then after you rounded, what did you do?

STUDENT: Well, um that's all I did.

ELYSHA PASSEGGI: Okay, so you started by rounding. Did anybody do it in a different way? Ryan?

STUDENT: Well, I looked at the relationship of the numbers.

ELYSHA PASSEGGI: Okay. And what did you notice?

STUDENT: And I noticed that from the forty-one to the forty-four you added three and from the thirty-two to the thirty-six you added four.

ELYSHA PASSEGGI: Thirty-two to thirty-six you added four. Okay.

STUDENT: They told me that one side must be bigger than another because the numbers... because they added more on one side than another.

ELYSHA PASSEGGI: Okay, so you're saying that by figuring out what was changing and noticing that this was a bigger number or bigger change that one side would be bigger than the other? Interesting. Does anybody have anything to add? Olivia?

STUDENT: Well that side being plus three and the other side being plus four, um I think it would mean that it would be false because they're two different numbers, so you can't cancel them out.

ELYSHA PASSEGGI: So since these are two different numbers, you can't just say we did the same thing on both sides? So you can't cancel that out? Did anybody see it in a different way besides plus three and plus four? Ellie what did you see?

STUDENT: I kind of did the same thing as Ryan but I know how to decide which one is bigger, which side is bigger. You could just add the one because you know that the forty and the thirty are going to be the same on each side, so whichever ones are higher, that's the higher.

ELYSHA PASSEGGI: That would definitely work but that is calculating. Did anybody see this differently though? I'm just curious, besides plus three and plus four, did anybody see it in a different way?

STUDENT: I did it like plus three and like minus four because thirty-six to thirty two is minus four.

ELYSHA PASSEGGI: So you saw it since you were changing from forty-one...what was changing from forty-one to forty-four and what was changing from thirty-six to thirty-two?

STUDENT: Yeah.

ELYSHA PASSEGGI: I'm just going to draw that underneath here. And what did you see changing?

STUDENT: Minus four.

ELYSHA PASSEGGI: Minus four. Did that change your opinion? Did you still see it as being false or?

STUDENT: Yes.

ELYSHA PASSEGGI: You did. And why did you see it as being false?

STUDENT: Because adding three and subtracting four, it doesn't (inaudible)

ELYSHA PASSEGGI: Okay, so the change is bigger here than here?

STUDENT: Yeah.

ELYSHA PASSEGGI: So does everyone agree that this one is false?

STUDENT: Yes.

ELYSHA PASSEGGI: Okay. Alright, one more. 29 + 56 = 27 + 58. Think about it for a second. Alright, does anybody have something? Maddy what do you see?

STUDENT: I think it's true because wouldn't you know that the tens are going to be the same? And if you add the nine and the six...well if you take one from the nine and added to the six then you get seven and eight. And that way you would know it's the same.

ELYSHA PASSEGGI: So this kind of strategy. If you trade one from here...if you trade just one... Is that what you're saying? Or trade something different than one?

STUDENT: If you trade, wait. If you do minus one from the twenty-nine and then plus one to the fifty-six then you get twenty-seven and fifty-eight, which is the same as the other side.

ELYSHA PASSEGGI: Okay, so twenty-nine take away one is giving you twenty-seven?

STUDENT: No, but if you take the nine and traded it, minus one, wait. If you take one from the nine and then you add the one to the six, you will get seven on the six and you'll get eight on the other one. Oh wait, never mind.

ELYSHA PASSEGGI: So if you were going to do that, what would you need to take from this twenty-nine to have it be the same on both sides? Two. And if you took two from here, you would get?

STUDENT: Fifty-eight.

ELYSHA PASSEGGI: Fifty-eight. So then it would work. So two would work. Did anybody look at this without doing calculations and just looking at the relationships of the numbers? David?

STUDENT: Well I saw the twenty-nine and fifty-eight. Well I just thought that it was true. I added twenty-nine and fifty-six and fifty-seven, that if you added, um... I just thought that it was true. I just guessed it.

ELYSHA PASSEGGI: Okay. Alright, Taniya?

STUDENT: Well I know that the twenty-nine and the fifty-eight were (inaudible).

ELYSHA PASSEGGI: You knew the twenty-nine and fifty-eight were what?

STUDENT: Well I just looked at the ones and I noticed that the eight was...well you had to add one to eight to get nine. And I just looked at the fifty-eight and twenty-nine.

ELYSHA PASSEGGI: So you noticed a relationship between these two numbers?

STUDENT: Not the tens but the ones, yeah.

ELYSHA PASSEGGI: Between these two numbers, the nine and the eight?

STUDENT: Yeah.

ELYSHA PASSEGGI: Okay. Did anybody notice a relationship between the numbers in a different way? I'm just curious because I'm hearing a lot of calculating, which will also give you the right answer. I'm just wondering if anybody noticed a relationship between these numbers. Rebecca?

STUDENT: I knew that twenty-nine was two more than twenty-seven.

ELYSHA PASSEGGI: Okay.

STUDENT: And fifty-six was two less than fifty-eight.

ELYSHA PASSEGGI: Okay. So, I'm just going to write this in a different way. Tell me if you think this makes sense. So you're saying that you had fifty-eight take away two equals fifty-six?

STUDENT: Yeah.

ELYSHA PASSEGGI: And that you had twenty-seven, sorry, twenty-seven plus two equals twenty-nine. So what would both the sides be equal to here? Is this true or false? Anybody know? Kate?

STUDENT: Since that's plus two and that's minus two then the other side would be plus two and minus two.

ELYSHA PASSEGGI: Okay, so you're saying because one is going up by two and one is going down by two then it's going to equal the sides because you're changing them in the opposite way? Which is what Heather was saying here, where it's canceling them out based on what we learned about hands on equations. Aden?

STUDENT: I noticed that the two numbers on the right side is fifty-eight and twenty-seven and then the answer is the other side, fifty-six and twenty-nine. I think they're true.

ELYSHA PASSEGGI: Let's try a challenge problem. Let's see if we can do a problem where I don't give you all the numbers and let's see if you can figure out what number is missing. You want to try that? Alright, what number would you put in this box to make the number sentence true? And remember no calculating. And I want you to think about what we talked about here. Frederick what do you think?

STUDENT: I think I would put a seven in the box because four is one less than five.

ELYSHA PASSEGGI: Okay.

STUDENT: Um, and eight is one more than seven. So they cancel each other out because they come with (inaudible).

ELYSHA PASSEGGI: Interesting. Did anybody else see it that way? Anybody disagree? Let's try another one. Twelve plus seven equals eight plus what (12 + 7 = 8 + __)? What do you think Rose?

STUDENT: Eleven.

ELYSHA PASSEGGI: Eleven? Derrick how did you come up with eleven?

STUDENT: Because eight is one more than seven.

ELYSHA PASSEGGI: Okay.

STUDENT: And then eleven is one less than twelve.

ELYSHA PASSEGGI: Ah. Is anybody noticing something? Is anybody noticing anything? Hannah what are you noticing?

STUDENT: That they're either going up or down one.

ELYSHA PASSEGGI: That they're either going up or down one? And that is the case for both of these? Does anybody notice anything else? Erick?

STUDENT: I notice that every time there's one arrow pointing to one side that it's always plus something. Then there's also another arrow pointing the other side that always says minus something.

ELYSHA PASSEGGI: Ellie?

STUDENT: Well I have an observation.

ELYSHA PASSEGGI: Uh-ha.

STUDENT: You did eleven going this way is minus one, it should be plus one.

ELYSHA PASSEGGI: Sorry you're right.

STUDENT: Yeah, and then also all the problems we've done are like something plus something equals something plus something.

ELYSHA PASSEGGI: Okay, past two, it's a number sentence (inaudible). Okay, Christopher?

STUDENT: Well each time we've done like one of those, well the first, um, seven is one less than eight and eight is one more than seven. So automatically I know that the twelve -- the one in the box is going to be one less than that.

ELYSHA PASSEGGI: Okay, interesting. Let's try the next one then and see if any of those theories work. Fifty-three plus what equals forty-nine plus thirty-five (53 + __ = 49 + 35)? What do you think? Does anybody have an idea on this one? I want you to take a second and talk to your shoulder partner and see what they think about this one.

STUDENT: I would think that since I made an observation for that there is one number on either side... If there's one number on one side and one number on the other side and this number is bigger, then this number has to be bigger. So I think that strategy...

ELYSHA PASSEGGI: Alright does anybody have an idea of what number is missing there? Anybody have an idea of what number is missing? Maddy what do you think?

STUDENT: Thirty-one.

ELYSHA PASSEGGI: Thirty-one? How did you guys come up with thirty-one? Oh here? So forty plus thirty is seventy?

STUDENT: Yeah. And nine and five is fourteen.

ELYSHA PASSEGGI: So you figured it out by doing calculations to figure it out?

STUDENT: Oh, I know.

ELYSHA PASSEGGI: You disagree?

STUDENT: Yes.

ELYSHA PASSEGGI: Okay, what did you do?

STUDENT: I did forty-nine is four less than fifty-three and so if it's four less, then you need four more for the thirty-one. The thirty-one is four more than the thirty-five and that's how I figured it out.

ELYSHA PASSEGGI: So you had to figure out what number plus four is going to give you thirty-five? Anybody else see it that way? Who saw it Maddy's way? One last problem. Eighteen plus twenty-four equals twenty-one plus what (18 + 24 = 21 + __)? Alright, what did you come up with? Quinn what did you come up with?

STUDENT: Twenty-one.

ELYSHA PASSEGGI: And how did you figure out twenty-one?

STUDENT: Well I took off three from the twenty-four and added it to the eighteen.

ELYSHA PASSEGGI: Oh, you did this. So that's what you did?

STUDENT: And then I have twenty-one and twenty-one.

ELYSHA PASSEGGI: How did you know this was twenty-one though?

STUDENT: Because three plus eighteen is twenty-one and twenty-four minus three is twenty-one.

ELYSHA PASSEGGI: Oh, so you wanted to get a twenty-one and a twenty-one on both sides that you could just see it. Did anybody see it in a different way, Olivia?

STUDENT: So first, I looked at the twenty-four and twenty-one.

ELYSHA PASSEGGI: These two?

STUDENT: Yeah. And it was minus three.

ELYSHA PASSEGGI: Okay, let me just cross that out.

STUDENT: And so from all the other problems it was...I was going to see if it was true or false. I made it plus three on the eighteen, plus three and that, like...it was equal sort of but...I don't really know exactly how to explain it.

ELYSHA PASSEGGI: It was equal sort of? Are you meaning the threes were the same?

STUDENT: Yeah. And then I looked at the eighteen and twenty-four and I kind of did what Quinn did to make it twenty-one and twenty-one.

ELYSHA PASSEGGI: I think that's an interesting observation. Alright, you guys did a really great job on these problems.

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classroom videos /number talks /4th grade math - can this be true? /number talk /

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Post Talk Number Talk ELYSHA PASSEGGI: So this morning we're going to do a number talk and we're going to talk about the same kind of ideas that we were talking about on Wednesday. So I want to just look back at what we discovered and talked about on Wednesday.

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Jo Teaching a Visual Dot Card Number Talk - YouCubed

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Uploaded by JSERoadrunners on 2013-10-04.

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Kit is an MfA Master Teacher teaching 6 th and 7 th grade math in a NYC public middle school. He is dedicated to crafting experiences for his students that create cognitive dissonance to develop students' mathematical mindsets. He meets students where they are, and challenges them to grow their brain and delve deeper into mathematical understanding. He is constantly reflecting on his own practice: sharing those reflections in his blog https://teachdomore.wordpress.com/ and tweeting at @MrKitMath
Challenging my Algorithm-Loving Students to Think
Recently I designed a sequence of strings to support my 7 th graders to reason about division of rational numbers. I wanted to move away from the "algorithm only/always" approach I had seen and help my students build a bank of smart strategies. Ultimately I am hoping that my students are flexible thinkers with deep number sense, so this set of strings was designed to explicitly invite them to try new, different strategies based on the relationships of the numbers in the problems.

In our investigation of division strategies, I launched our first number string by telling students to look for relationships they could use to make division easier. Our goal for the week was to think about when long division was necessary and when there were more efficient or better strategies that could be applied. Our first string was designed to have students notice the constant ratio - when both the dividend and divisor are multiplied or divided by the same constant - also known as scaling up or down down division problems.

32 ÷ 4

320 ÷ 40

3200 ÷ 400

3.2 ÷ .4

5.6 ÷ .8

Students noticed quickly that we could scale the problem up and down to make friendlier numbers - and that the quotient stayed the same. They described the division as a fraction, and related the scaling to simplifying fractions. They emphasized that it had to stay equivalent, but we could be flexible in changing the numbers. They were able to apply this strategy to the new problem without a helper.

Later in the week, I returned to the work and said the following, "This week, we've been working on division strategies, and considering what's the best, most efficient way to solve a problem. Today, we are going to do a number string with a bunch of division problems. Our focus is not going to be on speed, because it's not a race. Instead we are going to try to find the most efficient or easiest strategy to use. Our goal is to think like mathematicians - be strategic and efficient. Consider the numbers for each problems before you choose a strategy - and be ready to explain how your strategy makes the problem easier to do mentally."

For my first period class I planned this string

13.2 ÷ 1.1

3.6 ÷ 1.8

7.2 ÷ 4.5

245 ÷ 3.5

32 ÷ .25

3600 ÷ 1.4

(108 ÷ 2.4) - planned but didn't get to it

We did all of the problems, except the last one. Because of conversations we had earlier in the week, my students trusted that they could scale up or down a division problem to make it friendlier. This meant 13.2 ÷ 1.1 became 132 ÷ 11. My students knew that those two problems were equivalent. Though they were good at scaling by any power of ten, they did not take up the idea of scaling by other factors. Many of them got stuck on 7.2 ÷ 4.5 for example. They initially thought to scale the problem to 72 ÷ 45, but written as a division problem seemed not to help them. When it was written as a fraction, however [72/45] students knew they could rename it as 8/5 and later it became 1.6.

Next I posed the problem 245 ÷ 3.5 and students were not sure what to do. Interesting things happened:

A student scaled the problem to 2450 ÷ 35, then changed the problem to 2450 ÷ 70 (by doubling the divisor) to make it easier for him to think about. Next he used partial quotients to build his way up to the quotient - essentially 2100 ÷ 70 = 3 and 350 ÷ 35 = 10. Once he had the partial quotients for 2100 ÷ 70 and 350 ÷ 35, he multiplied 2 x 10 x 3 to get 60 35s in 2100 and 10 35s in 350, which he added together to get 70. I was struck by the power of his working memory to hold all of these parts together and knew that recording his strategy as he spoke it would help him, and all of my other students make sense of his thinking.

Another student scaled the problem inconsistently - 245 ÷ 5 became 2450 (scaled by 10) ÷ 350 (scaled by 100). Later the same student adjusted the problem by a factor of 10 to accommodate for the original move. I was fascinated by this strategy - adjusting the problem to make a non-equivalent, but friendly problem, and then adjusting it back to make it equivalent again.

The next problem - 32 ÷ .25 was surprisingly easy for the students to solve. I think they recognized .25 as 1/4 of a whole, whereas they do not think of 3.5 as 1/2 of 7. Many may have thought about money as well - envisioning the .25 as a literal quarter, four of which are equivalent to $1 and thinking about how many quarters in $32.

For the last problem - 3500 ÷ 1.4 - I was surprised by the number of my students who simplified the problem by a factor of 7 - 500 ÷ .2 and then "just knew" it would be 2500.

Finally, I borrowed a practice from the Contemplate then Calculate [ http://www.fosteringmathpractices.com/contemplate-then-calculate/] routine and asked students to reflect on their own thinking [meta-cognition]. I reminded them of our goal of thinking like mathematicians and finding new strategies for new problems. I allowed them time to choose a prompt and write a response on an index card that I collected.

After reading through their exit tickets it was clear that students were in many different places, with respect to division of decimals:

Some mentioned using "common factors" as a helpful strategy

Others mentioned noticing "common multiples" as a helpful strategy

Some mentioned "scaling up or down" to make the numbers friendlier/easier

Many noticed patterns but not all could describe them or say what was helpful about them

Several hadn't yet developed the language to describe the mathematics and wrote in vague terms - "having strategies that worked quickly"

For the next day, I planned an "entry slip" where students were asked to solve another decimal division problem using mental math and then ask them to record their strategies on an index card. I thought about using 108 ÷ 2.4, but initially worried it required too much scaling:

108 ÷ 2.4 = 1080 ÷ 24 = 540 ÷ 12 = 270 ÷ 6 = (240 ÷ 6) + (30 ÷ 6) = 40 + 5 = 45

Ultimately, I decide to try it out - not as a string, but as independent work, where students were asked to explain their thinking in words as well as in numbers. I asked student to "think about the strategies from this week's number strings and use them to solve today's problem."

When I analyzed their entry slips, their work fell into a few big categories:

Scaling the problem up and down until it feels friendly (Hiro)
Getting rid of the decimal (Ming) Looking for common factors in the numbers (12), then solving (Franklin) Scaling up the divisor to get rid of the decimal, then "making it equal" (Janice) Where are we now? Looking inside the numbers for common factors (12), then scaling up by 10 (Dante) Where am I now?
Multiplying the divisor (2.4) by 10 to get rid of the decimal, then adjusting (Wendy)

Initially, I saw many of my students struggle to solve division of decimals problems on a pre-assessment. So I was fascinated with how many of them were able to do our entry slip problem using strategies that had emerged in our strings. It's clear from the exit tickets, too, that most of the students were able to use the strategies, while a handful of them were resistant to leaving long division behind. Sure, there were some students who made some calculation errors, but this was true of those who used long division as well as those who scaled the problem to a friendlier place.

I am also thinking about how differently I lead number strings - from just a year ago. I know from other routines (Contemplate then Calculate) that very focused turn-and-talks at specific points in the routine is really important. I also watched Kara Imm ( Math in the City) do this with middle school students at Lyons Community School (Brooklyn) this fall - a way to give all kids an opportunity not just to think, but also to talk. My students had better stamina this year, and they were more interested in listening when I asked them to put their pens, calculators and notebooks away. Engagement was better both because of the structure of the routine as well as the way we designed the strings to build from one to the next.

By crafting an opportunity for students to see the efficiency of other strategies over the standard algorithm, I encouraged my students to move beyond the algorithm as a standard default. Now, instead of mindlessly attacking a problem with a brute force strategy such as long division, my students are beginning to think flexibly about other possible strategies. This is evident from the number of students whose exit tickets show no signs of long division!

I've found a few things are key in delivering a successful number string. First, the sequence of the problems needs to guide students towards specific strategies and expand their tool box one piece at a time, without narrowing their focus too much on one tool, such as when I accidentally blinded my students by providing them with too many scaling by 10 and not enough "obvious" scaling by other factors, such as 2, 3, 4, 5, or even 12! Second, though the number string is a whole-class activity, it can and should be broken up into chunks of partner talk where students discuss their strategies in their partnerships and then discuss the strategies that are shared out. Finally, the reflection component at the end of class is critical for ensuring that students learn strategies to solve future problems and not just one solution for one problem.

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Posts about division strings on Number Strings

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Kit is an MfA Master Teacher teaching 6 th and 7 th grade math in a NYC public middle school. He is dedicated to crafting experiences for his students that create cognitive dissonance to develop students' mathematical mindsets. He meets students where they are, and challenges them to grow their brain and delve deeper into mathematical understanding.

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How are number strings designed? Typically, people tend to describe number strings as having the following structure

Entry problem

Helper problems

Challenge problem (or clunker)

This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this:

Again, the concept of helper problems. But is there just one "formula" for a number string?

Many number strings have this helper problem structure, such as strings that develop jumps of ten (and multiples of ten) in addition and subtraction, like the number string above, as well as number strings that develop the use of the distributive property of multiplication such as:

Again, you can see the series of helper problems that scaffold increasingly difficult problems. But what about this string for early multiplication?

Here, the number string is designed with equivalent problem structure. The first four problems are all equivalent, allowing students to think through why doubling one factor and halving the other results in an equivalent product. Other number strings with equivalent problem structure are constant difference in subtraction strings, such as:

Here the first four problems are equivalent to illustrate how constant difference can make a seemingly difficult problem like 131 - 66 simpler to solve, by shifting the difference up the number line to 130 - 65.

There are also number strings that have a compare structure, such as this number string which asks students to consider whether to add on or remove in subtraction

This distinction matters when you are designing strings, and also when you are facilitating them. When you are facilitating the number string above with a compare structure, you want to make sure to draw out a discussion of which strategy works for which set of numbers, and why. Much like a compare discussion in Elham Kazemi and Allison Hintz's Intentional Talk, the structure of the mathematics leads to a particular structure for discussion. For a number string with an equivalency structure, you want students to be talking about the equivalency and proving that these problems are truly equivalent. For a number string with a helper problem structure, discussion can focus on which simpler problems helped you solve a more challenging problem. For all the problems, discussion can center around the patterns that they notice in the number string. But for different problem structures, those patterns are different.

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How are number strings designed? Typically, people tend to describe number strings as having the following structure Entry problem Helper problems Challenge problem (or clunker) This post from Math Coach on Demand (which also has a bunch of addition and subtraction number strings) describes the structure like this: Again, the concept of helper problems. But is...

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Coaching - It's teaching and learning from a different angle

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Coaching - It's teaching and learning from a different angle

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On Tuesday September 8, 2015 I was privileged to be the presenter on Global Math Department. I'm not sure that the Resources links that I had in my Powerpoint presentation work properly so I am reposting them here. Resources How to Start a 2 page document with facilitation tips.

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Discussion and Supporting Materials

Discussion

Supporting Materials

Thought starters

1. Why is it helpful for students to discuss the mental images they formed?

2. Ms. Latimer takes multiple student answers to the same question. What effect does this have?

3. How could this lesson be connected with addition and subtraction?

4. See also Mathematical Practice Standard 7 in the CCSS?

Newest

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Quick Images: Visualizing Number Combinations

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Stephanie Latimer begins her lesson on quick images by showing students an image of eight objects. She shows the image for only three seconds, which discourages students from counting by ones. Students put their thumbs up when they think they know how many objects are shown in the image. Stephanie c…

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Meeting Students' Needs in Number Talks Transcript

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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)

|

Math.5.NF.B.4

Common core State Standards

Math: Math

5: Grade 5

NF: Numbers & Operations--Fractions

B: Apply and extend previous understandings of multiplication and division

4:
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
<br />
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Download Common Core State Standards (PDF 1.2 MB)

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Discussion and Supporting Materials

Thought starters

1. How does Ms. Morey structure her class in order to provide tailored, small group instruction?

2. Why does Ms. Morey decide to use manipulatives with her small group?

3. How does Ms. Morey use this lesson as an opportunity to build her own skills?

School Details

Enumclaw Middle School
550 Semanski St
Enumclaw WA 98022

Data Provided By:

Teachers

Math / Kindergarten 1 2 3 4 6 / Coach

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Crystal Morey provides tailored instruction to a small group of students in her sixth grade math class. First, she formatively assesses all students. Next, she pulls together a small group of students who have similar misconceptions. In the small group, students use manipulatives and images to explore the math concepts.

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http://www.fractiontalks.com/#

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There has been an increase of activity on Twitter using Fraction Talks in the Which One Doesn't Belong? structure. Prompts like the...

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https://www.smore.com/8z161

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Making Number Talks Matter - Chapter 1 Newsletter by Kristin Gray | This newsletter was created with Smore, an online tool for creating beautiful newsletters for for individual educators, schools and districts