Illustrative Mathematics Algebra 1 Answer Key

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 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...

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...

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...

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...

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...

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...

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)...

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...

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...

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,...

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 +...

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...

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...

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....

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...

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...

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...