IM Certified® Blog

The Story of Grade 8 Unit 3: Linear Relationships

The Story of Grade 8 Unit 3: Linear Relationships

By Ashli Black and Elisa Smith Grade 8 is a year marked by shifts in mathematical focus. Where grades 6 and 7 introduce students to negative numbers and using them in operations, grade 8 introduces them to irrational...

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Creating Collaborative Math Classrooms

Creating Collaborative Math Classrooms

This blog post was written by William McCallum, co-founder and CEO of Illustrative Mathematics, and originally posted on the ImagineLearning blog. Collaboration is a core value at Illustrative Mathematics. Creating a high...

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The Story of Grade 4

The Story of Grade 4

By Patti Drawdy and Yenche Tioanda “Why not start the year with place value?” Kaneka Turner, Grade 4 Lead Writer, hears this question often. Isn’t making sense of and operating on large numbers pretty essential in grade 4?...

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The Story of Grade 3

The Story of Grade 3

By Mike Henderson As a former grade 3 teacher, I know first hand how daunting teaching math at this level can be. On top of developing fluency with addition and subtraction within 1,000, students need to learn about new...

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The Story of Grade 2

The Story of Grade 2

By Mike Henderson Teaching addition and subtraction in grade 2 is a challenging balancing act. As students move from numbers within 100 to numbers within 1,000, they need to use approaches that involve directly representing...

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Towards Coherence

Towards Coherence

By William McCallum Last week, we had our first large-scale in-person event in quite a while, a training for new and returning facilitators in Baltimore, with over 110 facilitators and 13 employees attending. I gave a...

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The Story of Kindergarten

The Story of Kindergarten

By Alexandra Clayton Many 5 year olds love to tell you (and show you) how high they can count. This skill of rote counting, saying the counting words in the same order each time, is a lot like singing the ABCs . While...

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Two Recommendations to Elevate Instruction

Two Recommendations to Elevate Instruction

By Kate Nowak, IM Vice President of Curriculum Development and Portfolio “The more intensely interested a teacher is in a student’s thinking, the more interested the student becomes in his or her own thinking.”—Eleanor...

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Math Language Routines: Discourse with a Purpose

Math Language Routines: Discourse with a Purpose

By Dr. Kristen Taylor, IM Certified® Facilitator  Math teachers can talk all day about math! We get super excited when we encounter someone else who enjoys these conversations as well. But too many of our students don't get...

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The Story of Grade 1

The Story of Grade 1

By Brianne Durst Grade 1 teachers have the awesome responsibility of introducing their students to, and helping them build an understanding of, the structure of our number system. This is no small task! Just think about how...

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A Thread Through Early Algebra 1

A Thread Through Early Algebra 1

By David Petersen, 6–12 Curriculum Specialist andAshli Black, Director, 6–12 Curriculum The Algebra 1 course in IM 9–12 Math™ begins with a unit on one-variable statistics, a choice that many find surprising. Why start with...

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Revisiting Distance Learning with IM K–12 Math™

Revisiting Distance Learning with IM K–12 Math™

With the surge of the Omicron variant, many schools are moving back to distance learning. Although we may not be excited to leave our classrooms again, we are better prepared for distance learning this time around. Most, if...

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Taking the Long View

Taking the Long View

By William McCallum A few weeks ago, my wife and I took a 17-day rafting trip through the Grand Canyon. We descended through the layers of rock, from the 270 million year old Kaibab sedimentary layer of chert, dolomite,...

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In the Beginning: Unit 1 in Kindergarten

In the Beginning: Unit 1 in Kindergarten

By Alex Clayton When was the last time you stepped into an entirely new environment? You weren’t sure exactly what was going to happen. There was no one you recognized. How did you feel: Uncertain? Anxious? Hopeful? During...

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Remembering Bob Moses

Remembering Bob Moses

By William McCallum and Kristin Umland "Math literacy will be a liberation tool for people trying to get out of poverty and the best hope for people trying not to get left behind.” —Bob Moses, 1935–2021 Bob Moses, the civil...

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Gifts: A Reflection on Student Thinking

Gifts: A Reflection on Student Thinking

By Elham Kazemi, Co-Author of Intentional Talk and Professor at University of Washington  “Gifts from the earth or from each other establish a particular relationship, an obligation of sorts to give, to receive, and to...

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Multi-grade Classrooms and  IM K–5 Math™

Multi-grade Classrooms and
IM K–5 Math™

By Jen Hawkins, IM Facilitator and IM K-5 Product Specialist Illustrative Mathematics believes that students can achieve success as mathematical thinkers by working through problems and consolidating their learning through...

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Introducing the IM Implementation Reflection Tool

Introducing the IM Implementation Reflection Tool

by Liz Ramirez, Director of Implementation “This makes the expectations for what I need to change visible.” “It’s not about the tool. It’s about the conversation using the tool.” Quotes from leaders who participated in IM’s...

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A Love for Math Reignited

A Love for Math Reignited

By Michael Ramirez, Senior Coordinator of School Transformation, Elementary Math When I began school as a kindergartener, I absolutely adored math. As a lower elementary school student, I remember relatives asking me what...

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IM K–5 Math: An End and a Beginning

IM K–5 Math: An End and a Beginning

By William McCallum On March 20, 2015, I received the following email: Thank you for submitting your proposals to the K–12 OER Collaborative. We are pleased to advise you that Illustrative Mathematics has been selected as a...

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Culturally Responsive Teaching and Math

Culturally Responsive Teaching and Math

By Asya Howlette, Director of Mathematics and Science at Thurgood Marshall Raise your hand if you have been perplexed by professional learning that told you your class needs to be culturally responsive, but left you...

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Differentiating Instruction with IM 6–12 Math™

Differentiating Instruction with
IM 6–12 Math™

By Max Ray-Riek, Director of 6–12 Professional Learning In my role at IM, working with teachers and administrators, I am asked to help with the challenges of implementing an IM curriculum. We are often asked, “How can I...

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What does it mean to enjoy mathematics?

What does it mean to enjoy mathematics?

By William McCallum When I started this series of blog posts on what it means to know, use, and enjoy mathematics, I thought this one would be the easiest. Math is fun, right? How could you not enjoy mathematics? I...

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The Joy of Fluency

The Joy of Fluency

By William McCallum I'm not a very good skier, but I have once or twice in my life experienced the joy of a perfect run: gliding down the slopes, taking each turn with confidence. And I've had terrible runs where I...

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A Circumference By Any Other Name…

A Circumference By Any Other Name…

By Becca Phillips You can’t tell a child “there’s no such thing as monsters” because, well, then, why is there a word for it? I was given this advice when my own daughter was an infant, but I’ve thought a lot about it over...

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What Does It Mean to Use Mathematics?

What Does It Mean to Use Mathematics?

By William McCallum Our vision at Illustrative Mathematics is a world where all learners know, use, and enjoy mathematics. In my last post I picked up that first verb and talked about what it means to know mathematics. In...

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Growing with the IM Community Hub

Growing with the IM Community Hub

By Portia Gibbs Roseboro, Britnee Wright, Justin Brennan The IM Community Hub, affectionately known as “The Hub,” was created to support educators using the IM Curriculum in navigating what teaching looks like now....

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What does it mean to know mathematics?

What does it mean to know mathematics?

By William McCallum A world where all learners know, use, and enjoy mathematics. Perhaps the most mysterious verb in the IM vision—a world where all learners know, use, and enjoy mathematics—is the first one: know.  Knowing...

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The Nuances of Understanding a Fraction as a Number

The Nuances of Understanding a Fraction as a Number

By Kristin Gray This was originally posted on Kristin Gray’s personal blog, Math Minds, on November 15, 2020. Student work is just the best. It is the one thing that will always motivate me to write! So, let’s kick this...

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Reading Graphs is a Complex Skill

Reading Graphs is a Complex Skill

by William McCallum Newspapers are full of graphs, far more than 10 or 20 years ago. Indeed, I have a graph to show that! (Source, Priceonomics) And yet I wonder how often readers see graphs as pictures illustrating a...

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Making Sense of Story Problems

Making Sense of Story Problems

by Deborah Peart, Grade 2 Lead Many people have an aversion to word problems. They cringe at the mention of them. In elementary classrooms, teachers often report that this is what their students struggle with most. When...

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The Story of Grade 5

The Story of Grade 5

by Sarah Caban From the start of the year, we want students to know they are capable of engaging in grade-level mathematics. In the Opportunity Myth (2018), data shows that there is an opportunity gap for historically...

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Planning for the Student Experience

Planning for the Student Experience

by Sarah Caban and Kristin Gray Teachers are so amazing and resilient. Amid all of the many thoughts and feelings about the challenges this school year brings, conversation continually revolves around their students.  When...

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Equitable Teaching Practices in IM 6–12 Math

Equitable Teaching Practices in IM 6–12 Math

by Tina Cardone The vision of Illustrative Mathematics is to create a world where learners know, use, and enjoy mathematics. This raises the question: Which learners? And what role do the authors of a curriculum play in...

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Coming Together Around Distance Learning

Coming Together Around Distance Learning

By William McCallum I can't imagine what it must feel like right now to be a teacher facing the uncharted territory that is the coming school year. Will I be teaching 100% online, or have some face-to-face interaction with...

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Thoughts on the Back-to-School Problem

Thoughts on the Back-to-School Problem

By William McCallum One of the consolations in these difficult times has been tweets and Youtube videos of parents discovering just what it takes to be a teacher. Maybe it takes a crisis like this to restore the respect...

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IM Talking Math

IM Talking Math

By Kristin Gray Most importantly, I hope everyone is taking care of themselves, their families, and others as much as they are able to during this time. With schools and districts pushing instruction online with a quick...

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Planning for Learning in Spring of 2020

Planning for Learning in Spring of 2020

Some schools are sending home printed packets and establishing teacher office hours by phone. Some are conducting their regular class schedule, but online. And lots are doing something in between. We understand that it is...

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Using math to make decisions in today’s pandemic

Using math to make decisions in today’s pandemic

By William McCallum At Illustrative Mathematics, our mission is to create a world where all learners know, use, and enjoy mathematics. This is not just an idle wish, one that we have because we love mathematics. Sometimes...

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Links to Resources for Shifting Instruction Online

Links to Resources for Shifting Instruction Online

First and most importantly, take care of yourself, your family, and your students. That might not look like doing math, or it might. To the extent that it’s useful, we have curated this list of resources recommended by our...

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Links to Math Resources for Caregivers

Links to Math Resources for Caregivers

Here is a collection of links the content team here at IM has used with our own students and kids to start mathematical conversations, play math games together, explore new topics, come up with projects, and have fun. There...

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When is a number line not a number line?

When is a number line not a number line?

By William McCallum The number line is a seemingly simple object: a straight line with two points marked 0 and 1. Those two points are the seeds of great complexity, however. Whole numbers are located at positions marked...

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The Art of Reflection

The Art of Reflection

“In times of stress, the best thing we can do for each other is to listen with our ears and our hearts and to be assured that our questions are just as important as our answers.” —Mr. (Fred) Rogers By Kaneka Turner We are...

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Ratio Tables are not Elementary

Ratio Tables are not Elementary

By William McCallum In grade 3, as students start to learn about multiplication, they think about products like 6 x 7 in terms of equal groups. 6 x 7 is the number of things when you have 6 groups with 7 things in each...

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Learning through Teaching

Learning through Teaching

By William McCallum I was in New Orleans a couple of weeks ago visiting a school using IM 6–8 Math and was inspired by the efforts the school was making to implement problem-based instruction. I saw teachers at different...

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Building a Math Community with IM K–5 Math

Building a Math Community with IM K–5 Math

“I’m not sure this is working. Only five of my students are participating and commenting each day. The rest sit there and look at me.” By Tabitha Eutsler This was my conversation with our math coordinator after my first few...

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Making Authentic Modeling Possible

Making Authentic Modeling Possible

The first thing you have to understand is that asking people to model with mathematics makes them mad. Not in all contexts, though! At a social gathering with a generally amiable and curious group of people, you might try...

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Which Vertex is the Center of a Triangle?

Which Vertex is the Center of a Triangle?

By William McCallum I am sometimes asked what is the secret to the success of our curriculum, what is the special property that sets it apart from other curricula. That question is like the one in the title of this blog...

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Building a Supportive Home/School Partnership

Building a Supportive Home/School Partnership

While families arrive with different school experiences and perspectives on what “doing math” means, they often share common questions: What do I need to know to set my child up for success in math this year? and How can I...

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Co-Creating Classroom Norms with Students

Co-Creating Classroom Norms with Students

Establishing norms is critical to creating an environment where all students see themselves as knowers and doers of mathematics. Reflecting on the Illustrative Mathematics mission statement, Creating a world where learners...

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First Impressions: The First Units in IM K–5 Math

First Impressions: The First Units in IM K–5 Math

“I've learned that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.” Maya Angelou  By Kristin Gray When I think back to my 8th grade math class, I...

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Introducing IM Certified 9–12 Math

Introducing IM Certified 9–12 Math

IM Algebra 1, Geometry, and Algebra 2 courses are now available to all. Alright, folks, this is not a drill: IM 9–12 Math is now available to all. By Ashli Black So now what? To help folks dive into the curriculum, we’ve...

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Inviting Students to the Mathematics

Inviting Students to the Mathematics

How do we invite students to the mathematics, and explicitly signal to kids that they have ideas that matter in math class? By Max Ray-Riek In this series of blog posts, the first of which is available here, we’re exploring...

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How Do Students Perceive Problem-Based Learning?

How Do Students Perceive Problem-Based Learning?

Does problem-based learning mean students need to forget everything they knew about how to act in math class? By Max Ray-Riek As a teacher, and then as a coach and teacher-educator, I’ve been thinking for a long time about...

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Making Peace with the Basics of Trigonometry

Making Peace with the Basics of Trigonometry

Six months ago, I hated trigonometry. By Becca Phillips In fact, when my daughter missed a week of school, she announced on her first day back, “Someone has to teach me trig because I missed the whole thing.” Her father...

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Realizing the promise of open resources, part II

Realizing the promise of open resources, part II

By William McCallum In my first post on the topic of realizing the promise of open educational resources, I described the IM Certified program. Our partners offer multiple versions, including a free online version and...

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Storytelling in the IM K-5 Math Curriculum

Storytelling in the IM K-5 Math Curriculum

By Kristin Gray, Director of K–5 Curriculum & Professional Learning Curriculum "An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and...

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The Power of Small Ideas

The Power of Small Ideas

By William McCallum, IM President Big ideas are popular in mathematics education, and you can find many lists of big ideas on the web. Some are more thoughtful than others, and I can see how some might be useful for...

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Making Sense of Distance in the Coordinate Plane

Making Sense of Distance in the Coordinate Plane

By Linda Richard, Curriculum Writer I used to teach my high school students a catchy song to memorize the distance formula. We all had fun goofily singing this song. My students hummed it to themselves during tests and...

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Designing Coherent Learning Experiences K-12

Designing Coherent Learning Experiences K-12

By Kristin Gray, Director of K–5 Curriculum & Professional Learning One challenge in curriculum design is considering all we know and believe to be true about math teaching and learning and translating that into...

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NCSM NCTM Recap

NCSM NCTM Recap

Illustrative Mathematics It was great to see so many of you at NCSM and NCTM in San Diego. If we missed you, or you weren’t able to attend, read our NCSM and NCTM round-up below. We enjoyed the conversations we had with...

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What is a Measurable Attribute?

What is a Measurable Attribute?

By Kristin Umland,VP Content Development A great conversation I had with the IM elementary school curriculum writing team got me thinking: What is a measurable attribute? That is, when given an object, what can we measure...

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Rigor in Proofs

Rigor in Proofs

By Tina Cardone, Geometry Lead, & Gabriel Rosenberg, Curriculum Writer There is no doubt that proof plays a central role in the human endeavor of mathematics, but there remains much debate on what role it should play in...

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Presenting IM Algebra 1, Geometry, Algebra 2

Presenting IM Algebra 1, Geometry, Algebra 2

By Kate Nowak When I was teaching high school mathematics, my local colleagues and I spent a whole lot of time creating problem-based lessons. We were convinced that this style of instruction was a good way to learn, but...

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How do you start the year?

How do you start the year?

By Ashli Black, Algebra 2 Lead Students need a chance at the beginning of the year to shake off the summer dust. Learn how IM's curricular design builds in opportunities for review while starting the year with inviting,...

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Planning Lessons for a Block Schedule

Planning Lessons for a Block Schedule

By Jennifer Wilson and Vanessa Cerrahoglu Update 2021 - August: IM has created block schedule guidance for IM 6-8 Math v.III and IM 9-12 Math v.I. Get unit guidance on how to customize the curriculum to fit your block...

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IM K-5 Math: Designing for Each Student

IM K-5 Math: Designing for Each Student

By Noelle Conforti Preszler and Kristin Gray In the following activity, think about the students in your classroom. How might each respond? What do you notice? What do you wonder? This activity is the drafted warm-up of the...

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What is problem-based instruction?

What is problem-based instruction?

By William McCallum When I was a child, I used to get puzzle books out of the library. One of the puzzles was the twelve-coin problem, the most difficult of all coin weighing problems. My mother and I worked on it...

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Extra Supports for Algebra 1: The Gateway Resources

Extra Supports for Algebra 1: The Gateway Resources

By Sadie Estrella Illustrative Mathematics’ high school curriculum is scheduled to be released this summer. This is an exciting time for Algebra 1, Geometry, and Algebra 2 teachers. I honestly am ready to take a job at a...

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Realizing the promise of open resources

Realizing the promise of open resources

By William McCallum All of our curriculum here at Illustrative Mathematics is released under a Creative Commons Attribution (CC-BY) license, which allows anyone to "copy and redistribute the material in any medium or...

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Truth and Consequences Revisited

Truth and Consequences Revisited

By William McCallum What are extraneous solutions? A while ago I wrote a blog post about solving equations where I talked about seeing the steps in solving equations as logical deductions. Thus the steps \begin{align*}3x +...

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What is the Time? It Depends…

What is the Time? It Depends…

Q: What is the fastest way to get a heated debate going about some topic in the IM 6–-8 math curriculum? A: Show people this graph from Lesson 4 in Unit 8.5: By Kristin Umland Many of us learned that time is always the...

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What is Multiplication?

What is Multiplication?

Multiplication is vexation, Division is as bad; The Rule of Three doth puzzle me, And Practice drives me mad. (old nursery rhyme.) Some people might answer that multiplication is repeated addition. For example, $5 \times 7$...

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The Power of Noticing and Wondering

The Power of Noticing and Wondering

My first years of teaching, I worried my students looked at me much like Ben Stein as the teacher in Ferris Bueller’s Day Off. I cringe to think about the series of monotonous and leading questions I strung together to a...

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Why is 3 – 5 = 3 + (-5)?

Why is 3 – 5 = 3 + (-5)?

By William McCallum You will never have to subtract again. Students sometimes learn about addition and subtraction of integers using integer chips. These are circular chips, with a yellow chip representing +1 and a red chip...

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Engaging All Students in Meaningful Mathematics

Engaging All Students in Meaningful Mathematics

“At the end of the day, this wasn’t about focusing on the objective, it was about making the objective meaningful to him.” The work of teaching is both invigorating and challenging. We want to instill a love of math and...

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Parent Math Night Using Illustrative Mathematics

Parent Math Night Using Illustrative Mathematics

Open House night; cue anxiety and sweaty palms! Hope my students’ parents don’t mind. I just began my seventh year of teaching middle school mathematics. Middle school is a limbo land filled with prepubescent pre-teens,...

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Planning for Meaningful Practice

Planning for Meaningful Practice

There is no shortage of available math resources for teachers to use in their classrooms. The difficult and time-consuming job for teachers is weeding through all of the tools to decide which best supports students in...

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Say What You Mean and Mean What You Say

Say What You Mean and Mean What You Say

By William McCallum In one of our professional development workshops, there is an activity in which the facilitator asks teachers to skip count by $\frac34$. The facilitator records the count, $\frac34$, $\frac64$,...

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What is right about wrong answers?

What is right about wrong answers?

When I first started teaching, at the end of each day, I would open my teacher’s guide, grab my pen, and thumb through the stack of completed worksheets. My eyes would dart quickly from the red answers in the teacher’s...

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What I Learned Today: Scale Drawings & Maps

What I Learned Today: Scale Drawings & Maps

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered,  “What did you learn at school today?” It took me a while to think about what I had learned (which will make me more...

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Planning to Use Pre-Unit Assessments

Planning to Use Pre-Unit Assessments

Time to start a new unit! What do you need to know before your students enter the room? NCTM’s Principles to Actions names several productive beliefs about assessments that will promote mathematical success for all. At the...

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IM Preparing for the School Year

IM Preparing for the School Year

There are always so many things to do in preparation for a new school year.  At this point of the summer, to-do lists start getting made, materials get purchased, rooms are organized, and math class planning begins....

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Building a Supportive Home/School Partnership

Building a Supportive Home/School Partnership

By Kristin Gray, Jenna Laib, Sarah Caban Open House. Back-to-School Night. Family Welcome. Math Night. No matter what the name of the event that launches the school year, family members will arrive at your school with the...

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Building a Mathematical Classroom Community

Building a Mathematical Classroom Community

Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms that expect students to communicate their mathematical thinking to their peers and...

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Fractions: Units and Equivalence

Fractions: Units and Equivalence

By William McCallum “I'm afraid I can't explain myself, sir. Because I am not myself, you see?” Alice in Wonderland. The idea of equivalence in mathematics is tricky for learners, because when we talk about two things being...

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5th Grade: Decimal Place Value

5th Grade: Decimal Place Value

By Kristin Gray There are some standards I think we do such a great job developing in early elementary, but never revisit explicitly when students learn about different numbers such as fractions and decimals. I blogged...

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