Illustrative Mathematics Geometry Unit 3 Answer Key

Using IM K–5 Math™ to Support Cognitively Guided Instruction (CGI)

Nov 16, 2021

By Mike Henderson "The activities require skills that students are not taught prior to doing the activity. They're expected to just jump right in. Why isn't there more modeling or re-teaching of basic skills so students can...

Remembering Bob Moses

Nov 3, 2021

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

Together Apart: Amplifying the "Gift of Student Thinking" in Distance Learning with IM

Oct 26, 2021

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...

Gifts: A Reflection on Student Thinking

Oct 12, 2021

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...

Unlocking Learners' Thinking Using the Mathematical Language Routines

Oct 5, 2021

By IM Certified Facilitators, Cheryl Fricchione and Rachel Rundstrom Consider this:A child whose primary language isn't English does not participate in the classroom to the degree that their English-speaking peers do....

FASTalk: Activating the Power of Families to Support Mathematics

Sep 28, 2021

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...

Choosing Math Tools and Representations that Work:

Choosing Math Tools and Representations that Work:
"100 chart-less in Grade 2"

Sep 20, 2021

By Maureen D. O'Connell, Math Specialist Some math tools and representations help students solve in-the-moment problems. Other math tools and representations help students solve problems and grow their conceptual...

Multi-grade Classrooms and IM K–5 Math™

Multi-grade Classrooms and
IM K–5 Math™

Sep 8, 2021

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...

Introducing the IM Implementation Reflection Tool

Aug 30, 2021

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...

Getting Ready for 2021–22: Back to School Blog Posts

Aug 10, 2021

By the IM Team Educators still face a great deal of uncertainty heading into the 2021–22 school year. Maybe you've already started planning for it. Maybe you've avoided planning for it. Maybe your school year has already...

A Love for Math Reignited

Jul 26, 2021

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...

IM K–5 Math: An End and a Beginning

Jul 13, 2021

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...

Culturally Responsive Teaching and Math

Jun 30, 2021

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...

Differentiating Instruction with IM 6–12 Math™

Jun 22, 2021

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...

Fluency Development Within and Across the Grades in IM K–5 Math™, part 4: Multiplication and Division

Jun 7, 2021

By Dionne Aminata, Grade 2–5 Lead Writer and Sarah Caban, Grade 5 Lead Writer"Some students are never given the opportunity to engage with mathematics in rich and meaningful ways that emphasize critical thinking and problem...

Fluency Development Within and Across the Grades in IM K–5 Math™, part 3: Multiplication and Division

Jun 1, 2021

By Dionne Aminata, Grade 2–5 Lead Writer and Sarah Caban, Grade 5 Lead Writer "The idea of being a 'math person' is based on two pervasive world views; first, the so-called 'fixed' mindset, that what one knows about math...

What does it mean to enjoy mathematics?

May 17, 2021

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...

Fluency Development Within and Across the Grades in IM K-5 Math™, part 2: Addition and Subtraction

Apr 27, 2021

By Sarah Caban, Grade 5 Lead Writer and Dionne Aminata, Grade 2–5 Lead Writer"We have to break from the notion that learning mathematics must be a linear and procedural endeavor mastered through rote practice and...

Fluency Development Within and Across the Grades in IM K–5 Math™, part 1: Addition and Subtraction

Apr 20, 2021

By Sarah Caban, Grade 5 Lead Writer and Dionne Aminata, Grade 2–5 Lead WriterWhen do students practice their math facts? How is math fluency assessed?"—current and prospective IM K-5 Math™ teachers everywhere We...

The Joy of Fluency

Apr 5, 2021

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...

A Circumference By Any Other Name…

Mar 31, 2021

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...

Building Equitable Learning Environments for Each Student

Mar 22, 2021

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...

Explorations in IM K–5 Math: Challenges for Curious Students

Mar 2, 2021

By Jen Hawkins and Mike Nakamaye What do we do with curious students who are ready, willing, and able to go further with math ideas? Some students cannot wait to dig deeper into the mathematics they are studying in class....

What Does It Mean to Use Mathematics?

Feb 22, 2021

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...

Supporting Culturally Responsive Pedagogy with IM K–5 Math™

Feb 17, 2021

By Dionne Aminata "We are striving to . . . compose a country committed to all cultures, colors, characters, and conditions of man. And so we lift our gazes not to what stands between us, but what stands before us." Poet...

IM 6–12 Math: Grading and Homework Policies and Practices

Feb 9, 2021

By Jennifer Willson 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 challenges is: how can we best align these...

By the End of Grade 3: Developing Fluency with Multiplication

Jan 25, 2021

By Zack Hill The major work of grade 3 includes representing and solving problems that involve multiplication and division. Then, by the end of grade 3, students are expected to know from memory all products of two...

Growing with the IM Community Hub

Jan 12, 2021

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....

What does it mean to know mathematics?

Jan 4, 2021

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...

Preparing for the Unknown: Our Journey to Virtual Facilitation

Dec 21, 2020

By Ashley Powell and Moniquea Willingham The excitement and nervousness in the room was almost palpable! There were approximately forty facilitators present for the 3-day IM K–5 Curriculum training in Dallas, Texas. We...

The Nuances of Understanding a Fraction as a Number

Dec 7, 2020

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...

Creating Time and Space for Students to Develop Foundational Mathematical Ideas

Nov 16, 2020

By Maureen D. O'Connell "Slow down, you're moving too fast, you got to make the morning last..." When we consider early childhood mathematics this familiar song comes to mind. In our hurried society where more is more,...

Reading Graphs is a Complex Skill

Nov 4, 2020

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 point, rather than as...

Making Sense of Story Problems

Oct 27, 2020

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...

The Story of Grade 5

Oct 15, 2020

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...

Using IM's Distance Learning Resources to Create a Hybrid Learning Plan

Sep 30, 2020

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...

Planning for the Student Experience

Sep 4, 2020

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...

Facilitating the

Facilitating the "Choral Counting" Routine Online

Aug 25, 2020

by Janaki Nagarajan How can we best do mathematics together in an online environment? When school suddenly shifted online last spring, I found myself overwhelmed by the learning curve for new technologies—for both myself...

Helping Elementary Students Cultivate a Strong Math Community

Aug 18, 2020

by LaToya Byrd and Jenna Laib School looks different this year. It's easy to focus on the changes that will need to be made—the new practices, the new routines, the new technologies—but we must first focus on our central...

Equitable Teaching Practices in IM 6–12 Math

Aug 11, 2020

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

Jul 27, 2020

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

Jul 15, 2020

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

Jul 7, 2020

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

Jun 25, 2020

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

Jun 15, 2020

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

Jun 8, 2020

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 to the Fall, Part 2: Creating a Supportive Resource for K–5 Teachers

May 20, 2020

By Kristin Gray, Director K–5 Curriculum and Professional Learningand Kevin Liner, IM K–5 Professional Learning Lead In our previous post, we highlighted important considerations in planning to support students in the fall....

Looking Ahead to 2020–21 in IM 6–8 Math and IM Algebra 1, Geometry, and Algebra 2

May 14, 2020

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...

Looking to the Fall, Part 1: Welcoming and Supporting K–5 Students

May 7, 2020

By Kristin Gray, Director K–5 Curriculum and Professional Learningand Kevin Liner, IM K–5 Professional Learning Lead It is overwhelming to think about how teaching and learning will look in the fall. The uncertainty of the...

English Learners and Distance Learning: Enhancing Access

May 3, 2020

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

Apr 27, 2020

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

Apr 8, 2020

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...

IM Talking Math

Mar 26, 2020

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...

Planning for Learning in Spring of 2020

Mar 23, 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...

Using math to make decisions in today's pandemic

Mar 16, 2020

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...

Aggregated Support for the IM Math Community in Spring 2020

Mar 14, 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

Mar 14, 2020

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

Mar 14, 2020

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...

Shifting Practices: Helping Everyone—from Students to Administration—Find their Voice in the Math Classroom

Feb 29, 2020

It was easy to say yes! By Crystal Magers Last spring, I was approached by our Math Coordinator and asked about piloting a new math program. I knew my staff was ready for building-wide consistency and we were ready to try...

K–5 Curriculum Design Features that Support Equity and Inclusion

Feb 29, 2020

By Dionne Aminata Before I joined the K–5 curriculum writing team at IM, I was a K–8 regional math content specialist for a public charter organization that largely consisted of Title I schools, or schools receiving federal...

Rethinking Instruction for Lasting Understanding: An Example

Feb 25, 2020

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...

When is a number line not a number line?

Feb 15, 2020

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...

The Art of Reflection

Feb 3, 2020

"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...

Ratio Tables are not Elementary

Jan 12, 2020

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...

Could you—or someone you know—be our newest IM Certified Facilitator? The Critical Role of IM Certified Facilitators.

Dec 8, 2019

"What I find distinguishes IM is that IM Certified Facilitators are uniquely supported by the IM authoring team to ensure the integrity of the curriculum remains intact." By Kiana Porter-Isom I was always interested in...

Using Diagrams to Build and Extend Student Understanding

Nov 25, 2019

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

Nov 21, 2019

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...

Learning through Teaching

Nov 10, 2019

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...

Building a Math Community with IM K–5 Math

Nov 2, 2019

"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...

Creating an Accessible Mathematical Community with IM K–5: the power of

Creating an Accessible Mathematical Community with IM K–5: the power of "yet" for students and adults

Oct 22, 2019

Does the perfect elementary math curriculum exist? Armed with a growth mindset and the Alpha IM K–5 curriculum, teachers in Ipswich Public Schools push their thinking to reach all mathematicians. By Maureen D. O'Connell I...

Using Instructional Routines to Inspire Deep Thinking

Oct 13, 2019

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,...

Making Authentic Modeling Possible

Oct 8, 2019

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...

Which Vertex is the Center of a Triangle?

Sep 23, 2019

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

Sep 15, 2019

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

Aug 30, 2019

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

Aug 25, 2019

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

Aug 2, 2019

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

Jul 29, 2019

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)...

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

Jul 23, 2019

"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...

Concrete Representations that Give Students a Way to Get Started

Jul 18, 2019

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...

Introducing IM Certified 9–12 Math

Jul 17, 2019

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...

Inviting Students to the Mathematics

Jul 8, 2019

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...

How Do Students Perceive Problem-Based Learning?

Jun 24, 2019

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...

Making Peace with the Basics of Trigonometry

Jun 17, 2019

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...

Realizing the promise of open resources, part II

Jun 17, 2019

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...

Storytelling in the IM K-5 Math Curriculum

Jun 4, 2019

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...

The Power of Small Ideas

May 21, 2019

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...

Making Sense of Distance in the Coordinate Plane

May 13, 2019

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...

Designing Coherent Learning Experiences K-12

May 7, 2019

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...

Developing Conceptual Understanding and Procedural Fluency

Apr 29, 2019

By Melissa Schumacher, Curriculum Writer Which is more important for students to have: conceptual understanding or procedural fluency? Does one have to be taught before the other can emerge? Some argue that procedure has to...

NCSM NCTM Recap

Apr 15, 2019

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?

Apr 8, 2019

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...

Rigor in Proofs

Mar 24, 2019

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...

Presenting IM Algebra 1, Geometry, Algebra 2

Mar 18, 2019

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...

How do you start the year?

Mar 11, 2019

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,...

Representing Subtraction of Signed Numbers: Can You Spot the Difference?

Mar 2, 2019

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

Feb 28, 2019

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...

IM K-5 Math: Designing for Each Student

Feb 26, 2019

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...

What is problem-based instruction?

Feb 19, 2019

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...

Extra Supports for Algebra 1: The Gateway Resources

Feb 3, 2019

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...

Realizing the promise of open resources

Jan 25, 2019

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...

Truth and Consequences Revisited

Jan 21, 2019

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 +...

What is the Time? It Depends…

Dec 18, 2018

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...

What is Multiplication?

Dec 11, 2018

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, $5 \times 7$...

The Power of Noticing and Wondering

Dec 3, 2018

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...

Catalyzing Change through the IM Algebra 1, Geometry, Algebra 2 Math

Nov 26, 2018

NCTM has called for structural and curricular changes in high school mathematics. Learn about how IM's high school curriculum is aligned with the vision put forth by NCTM to end tracking, implement effective targeted...

Proof in IM's High School Geometry (A Sneak Preview)

Nov 13, 2018

Supporting high-school students to write detailed, precise proofs is challenging. Learn about some of the design elements that IM used to invite students to a deep exploration of proof. In IM's high school Geometry...

Why is 3 – 5 = 3 + (-5)?

Oct 31, 2018

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...

Professional Learning Through a Fraction Task Progression

Oct 16, 2018

Teaching mathematics is a continuous cycle of identifying where each student is in their learning trajectory and determining meaningful ways in which to build on their current understandings. While we...

Engaging All Students in Meaningful Mathematics

Oct 10, 2018

"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

Oct 3, 2018

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

Sep 19, 2018

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...

Say What You Mean and Mean What You Say

Sep 11, 2018

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, $\frac34$, $\frac64$,...

What is right about wrong answers?

Sep 4, 2018

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

Aug 28, 2018

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

Aug 21, 2018

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

Aug 13, 2018

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

Aug 6, 2018

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

Jul 24, 2018

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

Jul 23, 2018

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...

Fractions: Units and Equivalence

Jul 10, 2018

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...

5th Grade: Decimal Place Value

Jun 26, 2018

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...

The Intersection of Fraction Talks and Clothesline Math: Formative Assessment and the 5 Practices

Jun 12, 2018

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

Jun 4, 2018

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

May 21, 2018

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...

$Untangling fractions, ratios, and quotients$

Untangling fractions, ratios, and quotients

May 14, 2018

By William McCallum In everyday language, $\frac{a}{b}$, $a\div b$, and $a : b$ are all different manifestations of a single fused notion. Here, for example are the mathematical definitions of fraction, quotient, and ratio...

On Similar Triangles

May 7, 2018

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

Apr 30, 2018

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

Apr 16, 2018

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...

Using Math Routines to Build Number Sense in First Grade

Apr 3, 2018

By Allison Van Voy When I started teaching four years ago, I had no idea how important number sense was to a student's math understanding. I was fresh out of college, brand new to teaching, and number sense was not a...

Sometimes the Real World Is Overrated: The Joy of Silly Applications

Mar 26, 2018

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....

Instructional Materials Matter: Interpreting Remainders in Division

Mar 20, 2018

By Jody Guarino We know instructional materials play a key role in student learning experiences but how do we ensure our students are learning from coherent high-quality instructional materials that engage them in critical...

Adapting Curriculum For Students to Know, Use and Enjoy Fractions

Mar 13, 2018

By Melissa Greenwald You know it is time for a change when half of the students in class are lost by the third lesson of a new unit. I teach third grade in a charter school in Philadelphia. We use Go Math! and each year I...

Learning Goals and Learning Targets

Mar 5, 2018

By Jennifer Wilson One of your students is asked, "What are you learning about today in class?" How does your student respond? "Nothing" "Math" "The questions on this worksheet" "Deciding if two figures are congruent"...

Warm-up Routines With a Purpose

Feb 27, 2018

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...

Adapting Problems to Elicit Student Thinking

Feb 18, 2018

By Jody Guarino As a teacher, I constantly wonder how I can elicit student thinking in order to gain insight into the current thinking of my students and leverage their thoughts and ideas to build mathematical...

A Fraction Unit Does Not Always Begin With Lesson 1

Feb 12, 2018

By Jared Gilman As I sat down at my local coffee shop to plan my upcoming 5th grade unit on fractions, a wave of dread spread across my body. I started having flashbacks to last winter, when my students' frustrations with...

Why is the graph of a linear function a straight line?

Feb 6, 2018

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

Feb 6, 2018

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...

The Illustrative Mathematics Team Reflect on the 5 Practices

Jan 30, 2018

The entire Illustrative Mathematics team spends a lot of time reading about teaching and learning. Most recently, we have been reading—some of us rereading—and reflecting on the 5 Practices for Orchestrating Productive...

Using the 5 Practices with Instructional Routines

Jan 22, 2018

By Robin Moore As a coach, how can I help teachers structure their lesson-planning in order for students to unpack their mathematical understandings? This question is always at the forefront of my mind as I reflect on my...

Vocabulary Decisions

Jan 18, 2018

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...

How the 5 Practices Changed my Instruction

Jan 16, 2018

By Alicia Farmer I am the type of teacher you want on your teaching team. I am the person that can remember vast amounts of details, predict potential obstacles, and meet any and all deadlines. My organized personality is...

The 5 Practices Framework: Explicit Planning vs Explicit Teaching

Jan 9, 2018

"Whether we're asking students to analyze a historical event, reflect on a text, or work toward a scientific discovery, we need to give students a chance to dig into the ideas on their own first." By Kristin Gray I've come...

Not all contexts have the same purpose

Jan 2, 2018

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...

Welcome to the new Illustrative Mathematics blog!

Dec 31, 2017

In continually moving forward with our vision of creating a world where learners know, use and enjoy mathematics, the Illustrative Mathematics team is so excited to announce the launch of our official blog! Our blog will be...

Info Gap Cards: The Hidden Gem

Dec 12, 2017

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...

Fraction & Decimal Number Lines

Dec 6, 2017

By Kristin Gray Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and...

Respecting the Intellectual Work of the Grade

Nov 30, 2017

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,...