Grades 6–8 Math
Coherence between Grade 8 and Algebra 1
By Courtney Ortega, IM Certified® Facilitator I was recently in a meeting where a participant declared, “Grade 8 and Algebra 1 basically have all the same standards.” Have you ever wondered this yourself? Have you heard...
Exploring the Lesson Synthesis: When do I actually teach?
By Lizzy Skousen, 6–12 Curriculum Writer, IM Certified® Facilitator During a problem-based lesson, the teacher does a lot of listening while monitoring student learning. When teachers are introduced to a problem-based...
Looking back at 2022: Year-End Closeout Message from Bill McCallum
2022 was a remarkable year of growth and achievement at IM. Our employees are energetic, inspired, and committed to our vision of a world where all learners know, use, and enjoy mathematics. I couldn't be more proud of what...
Help Students Know, Use, and Enjoy Quadratics
By Joseph Koelsch, IM Certified® Facilitator “I’ve said it before: equations are the devil’s sentences. The worst one is that quadratic equation, an infernal salad of numbers, letters, and symbols.” - Stephen Colbert Did...
Supporting Teachers During Implementation of Illustrative Mathematics: Big Ideas For Coaches and Teacher Leaders
By Sonja Twedt, IM Certified® Facilitator, 6–12 Special Projects Course Assistant Are you an instructional coach or teacher leader in a district that is adopting Illustrative Mathematics? If so, you’ve likely found yourself...
Exploring Ratio Reasoning with Student Work
By Tashana Howse, IM Certified® Facilitator The relationship between fractions, ratios, and proportions is introductory to students' development of the study of Algebra. Therefore, it is important for students to engage in...
Strategies for Instituting Equitable Math Instruction
Dionne Aminata, Senior Director of Strategic Initiatives, Marketing, Illustrative Mathematics Rolanda Baldwin, Vice President of Mathematics, UnboundEd Illustrative Mathematics and UnboundEd both agree that achieving racial...
Illustrative Mathematics and IM Certified Partners featured at NCSM and NCTM 2022
Members of the Illustrative Mathematics team recently attended two national conferences for math educators: NCSM (Sept 26–28, 2022) in Anaheim, CA, and NCTM (Sept 28–Oct 1, 2022) in Los Angeles. We enjoyed meeting so many...
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...
Tackling Wordy Problems: How the Three Reads Math Language Routine Supports Access for All Learners
by Joe Herbert “These problems are great, but they’re just so wordy. My students can’t handle all that reading!” Does this feel familiar? If so, you’re not alone! Many students and teachers struggle with the language...
The IM Community Hub is Becoming the IM Resource Hub—Here’s Why
“Any time we go to the next unit I'm looking for what we can use, tweak, modify, build off of. Everyone is always checking the hub—it has saved us so much time in terms of planning this year.” “It made my ability to support...
Creating a space for students to become expert learners
By Vanessa Cerrahoglu and Danielle Seabold, IM Certified® Facilitators Learners have brilliant mathematical ideas. How we teachers use IM 6–12 Math™ impacts how often students and teachers recognize this brilliance....
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...
IM Curriculum Provides Everything but the Kitchen Sink Needed for Lesson Planning
By Liza Bondurant, IM Certified® Facilitator/Associate Professor Do you spend hours hunting down materials for your lesson plans? How can you be sure the materials are aligned to your standards? How can you make your lesson...
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...
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...
Leveraging IM 6–12 Math Teacher Materials to Enhance Access to Grade-Level Mathematics
By Vanessa Cerrahoglu and Danielle Seabold, IM Certified® Facilitators Illustrative Mathematics (IM) believes each student has brilliant mathematical ideas has experiences and ideas that are valuable can make sense of and...
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...
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,...
What Does IM Think About Unfinished Learning and Supporting Students with a Variety of Entry Points? (Grades 6–12)
By the IM Team In a class of 25 students, there are 25 different learners with 25 different backgrounds. Students come into the classroom with varied experiences with school, with mathematical content, and with life. While...
Illustrative Mathematics Celebrates 10 Years: Letter from the CEO
2021 marks our 10th anniversary at Illustrative Mathematics, and we couldn’t be more proud to have impacted so many hundreds of thousands of students, teachers, and families since our humble beginnings in 2011. Our business...
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...
Together Apart: Amplifying the “Gift of Student Thinking” in Distance Learning with IM
By Maureen D. O’Connell, IM K–5 Math™ Pilot Teacher,IM Certified® Facilitator for IM K–5 Math “The more intensely interested a teacher is in a student’s thinking, the more interested the student becomes in his or her own...
FASTalk: Activating the Power of Families to Support Mathematics
By Elisabeth O’Bryon, co-founder and Chief Impact Officer at Family Engagement Lab and Megan Lorio, Managing Editor at Family Engagement Lab Engaging families in meaningful and equitable ways is a cornerstone of student...
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...
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...
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...
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...
Building Equitable Learning Environments for Each Student
By Danielle Seabold All students can be successful in mathematics. For most mathematics educators, we lean into this. We believe that all students can learn mathematics, that they can be successful. However, as we focus our...
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...
IM 6–12 Math: Grading and Homework Policies and Practices
By Jennifer Willson, Director, 6–12 Professional Learning Design In my role at IM, working with teachers and administrators, I am asked to help with the challenges of implementing an IM curriculum. One of the most common...
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...
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...
Using IM’s Distance Learning Resources to Create a Hybrid Learning Plan
By Lorie Banks Trying to plan for the 2020–2021 school year has been like trying to fly the airplane while building the wings. I am a career educator—a middle grades math teacher in an urban district in Western...
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...
English Learners and Distance Learning: Math Language Routines
by Vanessa Cerrahoglu, Jennifer Wilson, and Liz Ramirez We envision creating a world where learners know, use, and enjoy mathematics. Knowing and using math goes beyond calculating and evaluating. We create purposeful...
New IM 6–12 Resources for Addressing Unfinished Learning and Engaging Students in Distance Learning
by David Petersen and Kate Nowak In our previous post, we described how we are thinking about planning for next fall. We are also creating some new resources to support users of IM K–12 Math in the fall. Some of this is to...
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...
English Learners and Distance Learning: Compare and Connect
By Vanessa Cerrahoglu, Jennifer Wilson, and Liz Ramirez We envision creating a world where learners know, use, and enjoy mathematics. Knowing and using math goes beyond calculating and evaluating. We create purposeful...
English Learners and Distance Learning: Co-Craft Questions
By Jennifer Wilson and Liz Ramirez We envision creating a world where learners know, use, and enjoy mathematics. Knowing and using math goes beyond calculating and evaluating. We create purposeful opportunities for students...
English Learners and Distance Learning: Clarify, Critique, Correct
By Jennifer Wilson and Liz Ramirez We want to acknowledge that we are all in different situations that shape how we respond to the call to adapt our teaching to fit a model for distance learning. This impacts the access we...
Looking Ahead to 2020–21 in IM 6–8 Math and IM Algebra 1, Geometry, and Algebra 2
By David Petersen, Lead Curriculum Writer and Kate Nowak, Director of K–12 Curriculum Strategy This school year has been strange and stressful, and there is uncertainty about what next year will look like. Due to school...
English Learners and Distance Learning: Enhancing Access
By Liz Ramirez Which students are experiencing success in today’s “distance learning”? What barriers do other students face? While virtual learning platforms have made it possible for some live instruction to continue...
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...
IM Talking Math 6–8: Resources for Weekly Re-engagement
By IM 6–8 Math Team This week, IM is launching a new resource to support students and teachers with distance learning. Each week we will publish an open-ended prompt or image that invites math conversation, and a series of...
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...
Aggregated Support for the IM Math Community in Spring 2020
We want to share our deepest gratitude for the work each of you has been doing to protect yourselves, your families, your students, and your school communities, as you face hard decisions about how to support students while...
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...
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...
Rethinking Instruction for Lasting Understanding: An Example
By Kate Nowak How do we help our students build mathematical understandings that endure past the unit test? If we want students to construct strong, reliable bases of mathematical knowledge, our instruction needs to do more...
Using Diagrams to Build and Extend Student Understanding
By Jenna Laib and Kristin Gray Take a moment to think about the value of each expression below. $\frac{1}{4}\times \frac{1}{3}$ $\frac{1}{4}\times \frac{2}{3}$ $\frac{2}{4}\times \frac{2}{3}$ $\frac{3}{4}\times...
The 5 Practices: Looking at Differentiation Through a New Lens
By Catherine Castillo Our district had seen a downhill trend in standardized test scores in mathematics. This forced us, as educators, to take an intentional look at our teaching practices. The past few years have been an...
Using Instructional Routines to Inspire Deep Thinking
We want students to think about math deeply. Creatively. Analytically. Instead, what often happens is that students race towards quick solutions. So what can we do to support this other kind of thinking in class—the slow,...
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...
Updates to Supports for Students with Disabilities and English Language Learners in IM 6–8 Math
At Illustrative Mathematics we are committed to creating a world where learners know, use, and enjoy mathematics. We believe that every student can learn grade-level mathematics with the right opportunities and support. Our...
Preparing for the School Year, Updated with Tips for Staying on Pace
Last year, we put together some reading to help people get started planning their year with IM 6–8. Now, we have another year’s worth of blog posts to choose from, plus a shiny, new high school curriculum! So once again,...
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...
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...
Explicit Classroom Norms to Teach Kids How to Learn From Solving Problems
This blog post is the fourth in a series of four blog posts exploring the student experience of problem-based learning. The first three posts are available here: (1) “How Do Students Perceive Problem-Based Learning?” (2)...
Concrete Representations that Give Students a Way to Get Started
This blog post is the third in a series of four blog posts exploring the student experience of problem-based learning. The first two posts are available here: “How Do Students Perceive Problem-Based Learning?” and “Inviting...
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...
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...
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...
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...
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...
Representing Subtraction of Signed Numbers: Can You Spot the Difference?
By Greta Anderson & Patti Drawdy, IM Certified Facilitator I read the lesson three times through, but was still unsure why the number line below shows $3 - 7$. My aha moment arrived courtesy of the grade 1 standards....
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...
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...
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...
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...
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...
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...
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...
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,...
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...
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...
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...
The IM Curriculum Changed How I Think About Math Instruction
Growing up we usually think we are either a math person or not a math person. But, in preparing for this year I saw a picture that said ‘How to be a math person: Step 1: Do math Step 2: Be a person’ and I really started to...
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...
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....
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...
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...
The Intersection of Fraction Talks and Clothesline Math: Formative Assessment and the 5 Practices
By Jenna Laib My sixth graders are weary of pre-assessments. No matter how many times we discuss the goal of a pre-assessment–for me to learn more about their current strategies and understandings, so that I can design...
The IM 6–8 Math Curriculum Changed My Math Methods Experience
By Anna Polsgrove When I first started the Math Methods course at University of California, Irvine, all of my ideas on how to learn math took a complete 180. During the first two months, a million questions swirled in my...
Fun With Zooming Number Lines in Grade 8
By Charles Larrieu Casias The number line is an anchor representation that threads through the entire middle school curriculum. For this blog post, I want to focus on a creative use of the number line in grade 8 to explore...
On Similar Triangles
By Ashli Black The fact that a line has a well-defined slope—that the ratio between the rise and run for any two points on the line is always the same—depends on similar triangles. (p.12, 6–8 Progression on Expressions and...
NCSM and NCTM 2018 Roundup
It was great to see so many of you at NCSM and NCTM. 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 those of you that are using the IM 6–8...
Time to Noodle
By Kate Nowak This task is the first part of the culminating lesson of unit 2 in grade 8, which is about dilations and similarity. (You will need to create a free teacher account to open the link.) It is a variation on the...
Sometimes the Real World Is Overrated: The Joy of Silly Applications
By Charles Larrieu Casias One of the cool things about math is that it can provide powerful new ways of seeing the world. Just for fun, I want you to open up this lesson from the grade 8 student text. Take a quick skim....
Warm-up Routines With a Purpose
By Kristin Gray As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have...
Why is the graph of a linear function a straight line?
By William McCallum In my last post I wrote about the following standard, and mentioned that I could write a whole blog post about the first comma. 8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function,...
Why We Don’t Cross Multiply
By Kate Nowak (co-authored with Kristin Gray) “Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to...
Vocabulary Decisions
By Bowen Kerins A wide-ranging team worked together to develop the Illustrative Mathematics Grades 6–8 Math curriculum. Many of the authors were and are experienced teachers of Grades 6–8, while others are experienced high...
Not all contexts have the same purpose
By Nik Doran We sometimes use familiar contexts to understand new mathematical ideas, and sometimes we use familiar mathematical ideas to understand what is going on in a context. We do both of these things by looking for...
Info Gap Cards: The Hidden Gem
By Sadie Estrella May 2016 seems so long ago. I actually had to look it up on a calendar because I really thought it was more than 1.41666years ago. That was when I officially started this journey with Illustrative...
Respecting the Intellectual Work of the Grade
By Kate Nowak A thing that I think we did really well in Illustrative Mathematics 6–8 Math was attend carefully to really deep, important things that adults that already know math can easily overlook. For example,...
Assessment Principles in Illustrative Mathematics 6-8 Math
By Bowen Kerins A wide-ranging team worked together to develop the Illustrative Mathematics Grades 6-8 Math curriculum. As Assessment Lead, it was my responsibility to write and curate the Shared Understandings...
Reflection & Discussions in Grade 8, Part 1
By Ashli Black Woo, blogging! As I start work on high school curriculum, I thought I would go back and revisit the grade 8 units that I’ve spent the past 18 months working on and share some of my favorite things. This gives...
Fraction division part I: How do you know when it is division?
By William McCallum and Kristin Umland In her book Knowing and Teaching Elementary Mathematics, Liping Ma wrote about this question and how teachers responded to it: Write a story problem for $1 ¾ \div ½$. [Pause here and...
Truth and consequences: talking about solving equations
By William McCallum The language we use when we talk about solving equations can be a bit of a minefield. It seems obvious to talk about an equation such as $3x + 2 = x + 5$ as saying that $3x+2$ is equal to $x + 5$, and...
Misconceptions about Multiple Methods
By William McCallum You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the...
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