# Grades 3–5 Math

## 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...

## 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

“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, childhood expectations have...

## 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...

## 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 “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 K–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...

## 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: 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...

## 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 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...

## 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

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...

## 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...

## 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...

## 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...

## 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 “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,...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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....

## 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...

## 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...

## 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...

## 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...

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

## 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....

## 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 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## Ways of thinking and ways of doing

Jun 7, 2017

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

May 24, 2017

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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#### RECENT POSTS

- Growing with the IM Community Hub January 12, 2021
- What does it mean to know mathematics? January 4, 2021
- Preparing for the Unknown: Our Journey to Virtual Facilitation December 21, 2020
- The Nuances of Understanding a Fraction as a Number December 7, 2020
- Creating Time and Space for Students to Develop Foundational Mathematical Ideas November 16, 2020

#### Categories

- Grades 3–5 (56)
- Grades 6–8 (70)
- Grades 9–12 (48)
- Grades K–2 (36)
- Professional Learning (39)