**IM Certified**^{®} Blog

^{®}Blog

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## Developing Conceptual Understanding and Procedural Fluency

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

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...

## 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...

## 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...

## 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,...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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 +...

## 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...

## 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, $5 \times 7$...

## 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...

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

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)

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)?

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

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

“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...

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

## 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...

## 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...

## 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...

## 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...

## Untangling fractions, ratios, and quotients

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

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...

## What is an instructional routine?

By William McCallum and Kate Nowak People use routines for all kinds of things. Routines give structure to time and interactions. People like structure. When a child comes home from school, there might be a routine. She...

## Using Math Routines to Build Number Sense in First Grade

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

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

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

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

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

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

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

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?

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...

## The Illustrative Mathematics Team Reflect on the 5 Practices

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

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

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

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

“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

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!

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

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

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

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...

## Ways of thinking and ways of doing

By William McCallum Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this: $$ 6 \times 3 = 6 + 6 + 6 = 18. $$...

## 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...

## The Structure is the Standards

Co-authored by Bill McCallum, Jason Zimba, Phil Daro You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will...