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ILIM K-5 Statement on Practice and Homework.pdf

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Making IM Centers Work: Joyful Practice, Meaningful Fluency, and Authentic Assessment - IM CERTIFIED® BLOG

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By Maureen D. O’Connell “How is it going with your implementation of IM K–5 Math Centers?” That’s my favorite question to ask during professional learning for IM K–5 Math. The response is a pedagogical temperature check about where teachers and grade levels are in their IM journey. Schools that repo…

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Video: Supporting Math Learning Outside of the Classroom

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Video: Supporting Math Learning Outside of the Classroom

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Supporting Math Learning Outside the Classroom

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Offline for maintenance 01/24 @ 9:00pm ET

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Each unit provides a family support video which includes prior learning, vocabulary explanations, math in pictures, and an easy-to-implement activity all designed to help families and caregivers successfully support their student with math learning at home. These videos also have closed captioning in Spanish.

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Family support materials, such as IM's family support videos, can be shared with families by sharing the video or resource URL without your site's subdomain.
Example: https://hogwarts.ilclassroom.com/lesson_plans/232834-4-1-unit-launch-family-support-video.

Remove the "hogwarts." and start with ilclassroom. This will take families to the public IL Classroom site where they don't need to be authenticated.

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Watch this training to learn more about the ILC Platform Navigation

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Watch this training to learn more about the ILC Platform Navigation

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Getting Started with Imagine IM K–5 - Getting Started - IL Customer Learning Library

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Take a deep-dive into learning more about problem-based learning, available resources to get the most out of the curriculum and our supportive community for Imagine IM.

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Imagine IM: Enhanced Illustrative Mathematics

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Guidance for using IM Kindergarten Math with Younger Students

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Guidance for using IM Kindergarten Math with Younger Students

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IM Author Bill McCallum paints a picture of what Illustrative Mathematics will look like in your classroom.

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By William McCallum, IM CEO
Here at IM we are excited by and proud of the reception IM K-12 Math™ is getting. As more and more districts adopt it, we are thinking about how to support them. What are the next steps? Our vision of a classroom community informs this work. We picture students actively engaged with and excited by the materials, confidently persevering in solving problems and sharing their thinking. We envision the teacher leading productive and meaningful discussions that help synthesize the students' learning of rigorous grade-level content. How can we make this vision a reality in more schools?

A coherent curriculum , like IM K-12 Math™, is critical to building classroom community, but more is needed to support teachers, schools, and districts with instructional shifts that promote access and equity for all students. There should be a shared vision of what the classroom will look like, and consideration of the support teachers will need. At IM, we call our vision the IM Classroom.

What is the IM Classroom?

IM's work is anchored in the belief that all students are capable of learning grade-level mathematics. Our curriculum supports a problem-based instructional model, where students are invited to the mathematics through relevant and affirming contexts. This draws on their naturally inquisitive nature, and their experience with the world around them. Problems provide multiple access points and solution paths, making them accessible to students at different points in their learning journey.

Problem-based instruction means believing all students can solve problems on their own and giving them a chance to try. This sets them on a path to developing mathematical identities as learners who can know, use, and enjoy mathematics. When students are engaged in their learning, they develop stronger perseverance and better retention of knowledge.

Four Elements of the IM Classroom

1. Teachers use an IM Certified® curriculum. practice IM's problem-based instructional model with integrity, and believe that all of their students are capable of learning grade-level mathematics.

2. Teachers participate in IM Certified Professional Learning, and have access to implementation support.

3. School leaders and administrators understand and support the systemic changes that are necessary to change the teachers' practice.

4. Families and communities engage with and support their students' learning.

Let's take a closer look at each pillar.

Pillar 1: Implementing problem-based instruction with IM Certified curriculum

IM Classroom teachers are passionate about learning and implementing our problem-based instructional model with integrity. They make sure their students understand each problem, monitor their students' work and advance their thinking with probing questions, provide information as appropriate, and, importantly, synthesize their students' learning at the end to make sure the goals of the activity have been met.

In an IM Classroom, the curriculum is obtained from one of our IM Certified Partners or by accessing the free, open educational resource (OER) on our website. IM Certification guarantees alignment with our coherent progressions of concepts and procedures. IM certified solutions include print and online tools, such as instructional videos and digital centers and manipulatives that support teachers to meet the needs of all learners.

Pillar 2: Teachers participate in IM Certified Professional Learning

When professional learning is certified by Illustrative Mathematics, it means that high- quality curriculum-aligned content is delivered by qualified facilitators who undergo rigorous training and ongoing development. IM Certified Professional Learning provides a coherent suite of offerings, both onsite and virtual, that support teachers and leaders through initial adoption and ongoing learning.

IM Certified Professional Learning Facilitators are experts in IM K-12 Math™ and the problem-based instructional model. In the IM Classroom, teachers continue to grow in both their practice and their understanding of the curriculum. Their participation in ongoing professional learning supports these efforts.

Pillar 3: School leaders and administrators understand and support necessary changes

While teachers have a direct impact on students' learning experiences, school and district leaders play an important role in making sure teachers have the appropriate tools and resources for their students to thrive. For successful implementation of IM K-12 Math™, leaders have a deep understanding of the problem-based instructional model, pedagogical practices, and planning protocols in order to create systems of support.

The IM Classroom exists in a building where school leaders and administrators support teachers and create systemic change through coaching, collaborative planning, implementation, and long-term support. IM Certified Professional Learning offers products and services for building leaders in order to support this effort.

Pillar 4: Families and communities engage with and support student learning

Families are students' first teachers. At IM, we believe that students should be empowered to recognize and explore math in the world around them, whether they are at school, at home, or in their communities. IM curricular materials include family supports that provide an overview of each unit and interactive mathematical questions that families can ask their children as the unit progresses.

IM is also developing relationships with organizations such as Family Engagement Lab, whose (Families and Schools Talk) tool includes families as instructional partners by providing curriculum-aligned learning activities to families via text message in their home language. The IM Classroom honors caregivers as critical stakeholders, which moves everyone towards a collaborative and supportive learning experience.

What happens in an IM Classroom?

When a student enters an IM Classroom, they enter a space that will encourage participation, develop confidence, inspire collaboration, and most importantly, equip them to know, use, and enjoy mathematics. The essence of the IM Classroom is the idea that learning happens when you pay attention to all of the actors-the students, the teachers, and the mathematics (yes, mathematics is alive in the IM Classroom!)-and build an instructional model that supports the complex community that they form.

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Beyond Curriculum Adoption: A Vision of the IM Classroom - IM CERTIFIED® BLOG

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By William McCallum, IM CEO Here at IM we are excited by and proud of the reception IM K-12 Math™ is getting. As more and more districts adopt it, we are thinking about how to support them. What are the next steps? Our vision of a classroom community informs this work.

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Each unit includes a guide that explains the key ideas and concepts in family-friendly language.

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Unit Assessments, Digital Task Statements, Cool-Downs, etc.

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Assigning, scheduling, changing assignment status, grading, feedback, sorting by students' first and last name, moving assignments to a different class.

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The 5 practices were designed to help teachers use

students’ responses to advance the mathematical

understanding of the class as a whole.

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There are 3 major design principles to support all learners: provide access to all, presume competence, provide a strength-based approach.

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Illustrative Mathematics with Imagine Learning is a dynamic, engaging instructional experience that leverages the power of high-quality curricula.

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Consider what your students need to encounter math more deeply and in collaboration with others. More heads are better than one!

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Math teachers can talk all day about math! We get super excited when we encounter someone else who enjoys these conversations as well.

But too many of our students don't get to experience that joyful discourse, which limits their understanding of and positive experiences with math.

How would our students' math learning and perception about math improve if they also had regular opportunities to talk about math? How might these opportunities help us create a world where learners know, use, and enjoy mathematics?

Let's explore the Math Language Routines, which are embedded in IM K-12 Math.

What are math language routines?

Stanford Graduate School of Education's Understanding Language and the Stanford Center for Assessment, Learning, and Equity (SCALE) designed eight Math Language Routines (MLRs) to help teachers "amplify, assess, and develop students' language" in math class. These routines are structured, but adaptable to the content and classroom context. The routines "emphasize the use of language that is meaningful and purposeful, not inauthentic or simply answer-based . . . [and] can and should be used to support all students learning mathematics." (from Understanding Language/SCALE's Principles for the Design of Mathematics Curricula: Promoting Language and Content Development ) There are four specific design principles used to entice students to discuss the concepts and ensure that the discussions provide opportunities for students to learn something new. The Math Language Routines also provide opportunities for students to engage in the Common Core's Standards for Mathematical Practice .

As students engage in math language routines, MLRs help students to advance their thinking in the following ways:

OPTIMIZING OUTPUT as students describe their own math thinking to others orally and visually or writing as physical, graphical, pictorial, or symbolic representations.

SUPPORT SENSE-MAKING as students scaffold their thinking and amplify language so students can make their own meaning between concepts and representations.

CULTIVATE CONVERSATION by providing opportunities to simultaneously make meaning, communicate that meaning, and refine the way content understandings are communicated.

MAXIMIZE META-AWARENESS as students organize their own experiences, ideas, and learning for themselves through reflection of the experience of discussing their math ideas, reasoning, and language.

This doesn't just mean that teachers tell students to talk about math and they do. We know that if we simply open the space and tell students to talk, much of the discussion will veer off-topic. Math Language Routines help teachers create a structure in which students can bring those four design principles to life.

Let's see what opportunities each MLR provides for students to practice exploring their math thinking and the MLRs' connections to the Standards for Mathematical Practice .

MLR 1: Stronger and Clearer Each Time - Students write a response to a math problem, then verbally share their response with a partner to get feedback from the listener to improve the response, and revise their original written response based on the feedback they received.

Students learn to both listen to their own thoughts and the thoughts of another with the purpose of strengthening their "first draft" thinking. This MLR strengthens student communication of their thinking. They make their explanations clearer through repeated "reviews" and revisions, which allow them to practice giving, receiving, and applying constructive feedback, and develop listening skills. This MLR helps students communicate their mathematical reasoning with greater clarity and focus.
(This connects to Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision. As students verbalize and review in writing the strategies they used and justification of those strategies, students have the opportunity to get feedback which can be used to critique the precision of their solution strategy and answer.)
MLR 2: Collect and Display - Students access their own and others mathematical ideas as the teacher scribes the language, strategies, and concepts students use during partner, small group, or whole-class discussions using written words, diagrams, and pictures.

By creating stable representations of student thoughts, the teacher extends the involvement of students in the math classroom. Student contributions are displayed for everyone to see. During that discussion as well as future conversations, students refer to these written ideas to improve the clarity and focus of their own communication. They can use the vocabulary, reference previous relationships between concepts, and identify new connections. This MLR helps make student thinking visible and promotes student ownership over mathematical ideas.
(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others, and 6: Attend to precision. As students view the thinking of their classmates from the representation, they can address any differences they see between their representation and their classmates' representation in type and kind, further pushing students to find the most efficient or precise representation to solve the problem.)
MLR 3: Critique, Correct, and Clarify - Students rewrite a math response from an example that is incorrect, incomplete, or otherwise ambiguous.

In addition to providing students with a space to further engage in math, students employ a "growth mindset" and consider revisions to their work in order to strengthen their communication. This MLR helps students "try on" others' math thinking in order to improve it.
(Standards for Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Similar to doing error analysis, students have the opportunity to engage with a strategy for solving that might be different from theirs in order to find ways to improve the situation representation, address and clarify misconceptions about the math, and "borrow" parts of the strategy that were relevant to add to their toolbox.)
MLR 4: Information Gap - In a group, each student has different parts of the math context and they work together to piece together that information orally and/or visually to bridge the gap between the parameters of a math situation and a question to solve a math problem.

As students work to make sense of individual pieces of information about a math context, they also identify what information they need in order to answer the math question. This creates an environment where students work together as they consider which strategies are the most efficient and what information they need to know to solve the problem. This MLR helps students explore math contexts with a purpose.
(Standards for Mathematical Practice 2: Reason abstractly and quantitatively, and 4: Model with mathematics. As students imagine the kinds of information they need to know and explain why they need to know that information, they are building an understanding of the problem and possible solutions related to the parameters of the problem and the kinds of data they need to solve for an answer.)
MLR 5: Co-Craft Questions and Problems - Students take a provided math "answer" to create a problem that could result in that answer or use a given context to create a problem that can be answered using that context.

With this MLR, students use their creativity to explore a math context as they iterate their understanding in order to identify questions that can be asked and answered by the context. As students share their understandings with others, they also uncover the importance of slowing down to explore and think before they try to answer a question, which results in students understanding the math more deeply. This MLR helps students define math contexts.
(Standards for Mathematical Practice 4: Model with mathematics. By creating their own questions and problems, students learn to visualize the parameters of the problem as they consider what kind of answers are possible with the information the problem provides.)
MLR 6: Three Reads - Students are guided to read the problem three separate times with three separate purposes with quick discussions between each read.

When students slow down to consider how they can use the information in a problem, they notice things about their understanding of the problem context and their own math understanding that they wouldn't have noticed before. Giving students a purpose to read helps to focus students to identify (1) the context of the mathematical situation, (2) relationships that exist within that context, and (3) strategies that can be used to answer the question using the problem context. This MLR helps students think metacognitively about their math problem solving.
(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them. As students set a purpose for reading each of the three times, students practice and see the importance of understanding the problem context and using the information they know from the problem in order to identify a strategy for solving that relates directly to the problem context and available information.)
MLR 7: Compare and Connect - Students identify, compare, and contrast their own understandings with other students' mathematical approaches, representations, concepts, examples, and language.

As students encounter other's math thinking, this MLR opens the floor for students to consider how they see and talk about math. This MLR specifically helps students interact with each other's thinking.
(Standards for Mathematical Practice 4: Model with mathematics, 6: Attend to precision, and 7: Look for and make use of structure. As students compare (find differences) and connect (find similarities) between their strategy for solving and another classmate's strategy for solving, students mine their understanding of the math concept in relation to the structure and steps of another student's understanding of the math. This allows both students to attend to precision because to be understood by another, they have to be precise with their explanation.)
MLR 8: Discussion Supports - Students access strategies in multi-modal ways, like visuals, aural, animation/movement, and symbols as modes, to help students better make sense of complex language, ideas, and classroom communication.

Enhancing access to the mathematical context through these multi-modal supports helps students develop understanding while also continuing to work on their mathematical communication. This MLR helps students deepen their mathematical understandings, and find ways to communicate their understandings in different ways.
(Standards for Mathematical Practice 1: Make sense of problems and persevere in solving them, 3: Construct viable arguments and critique the reasoning of others, 5: Use appropriate tools strategically, and 6: Attend to precision. Using discussion supports helps students to see the reasoning of others as well as identify which tools to solve are most appropriate. In using these supports, students can be successful with in-depth or multistep problems with support from teachers and classmates.)
Next Steps

Consider what your students need to encounter math more deeply and in collaboration with others. More heads are better than one!

Looking for more support for distance learning ? Check out IM's collection of resources.

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Math Language Routines: Discourse with a Purpose - IM CERTIFIED® BLOG

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Building fluency through centers by demonstrating flexibility, accuracy, and efficiency.

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Eight Math Language Routines (MLRs) are embedded into lessons to support in both learning the English language and learning the language of mathematics.

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The beginning of each unit includes an Inspire Math Video which showcases the mathematics of the unit in a real-world, engaging context. These videos are also available in Spanish.

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The instructional routines embedded within the Imagine Learning Illustrative Mathematics lessons create structures

so that all students can engage in and contribute to mathematical conversations.

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Within the Imagine Learning Classroom there are many types of virtual manipulatives that can be accessed in multiple places and in various lessons.

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The instructional design of the materials supports all learners through a coherent progression of the mathematics based

on the standards and research-based learning trajectories.

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Blog: Using Instructional Routines to Inspire Deep Thinking

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Blog: Using Instructional Routines to Inspire Deep Thinking

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Using Instructional Routines to Inspire Deep Thinking - IM CERTIFIED® BLOG

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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, deep kind? By Jenna Laib One way is through instructional routines like “Which…

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K–5 Unit Adaptation Overview | IL Classroom

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Explains the beliefs and designs behind the creation of unit adaptation packs for grades K–5. [Updated for Imagine IM]

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Blog: Fostering Teaching Practices to Nurture Opportunities for Students to Shine

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Blog: Fostering Teaching Practices to Nurture Opportunities for Students to Shine

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Fostering Teaching Practices to Nurture Opportunities for Students to Shine - IM CERTIFIED® BLOG

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By Tori Cole, Specialist, IM Certified Facilitator Timesha Brooks, Specialist, Math Client Support Engaging Reluctant Math Students Have you encountered students who are reluctant to engage in math? Students who have already decided they are “bad” at math? How do we support students whose mathematical identities seem fixed before they even enter our classrooms? As […]

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Blog: Annotate and Acknowledge Student Thinking

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Blog: Annotate and Acknowledge Student Thinking

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By Jen Hawkins, IM Certified® Facilitator and Independent Curriculum Implementation Specialist

As I sat at the table in the back of the room, I watched the teacher reveal an image connected to the lesson's warm-up. She told her students they would have a minute to consider the image, and then she would ask them about their thinking. Her students gave a thumbs-up when they had an idea to share. After waiting a true minute, she selected a student to share their idea about the image.

She wrote down a couple of words that the first student said. Faces brightened. Eyes lit up. The students' excited expressions revealed what they were thinking: "The teacher thought this was something interesting! Maybe even something important!"

The teacher called on another student who was waiting expectantly with a thumbs-up signal. She circled the part of the image that the student referred to, and the student smiled broadly. Students on the carpet looked eager to join the mathematical discourse happening around them. I could tell students' ideas were acknowledged and appreciated in this classroom. I was observing a vibrant, thinking math community.

Annotate & Acknowledge

How you launch math class matters. Starting an Illustrative Mathematics lesson with all students engaged and tuned into the same image, expression, or equation promotes a thriving math community.

We want students to feel secure and begin class from a place of assurance. Listening to all ideas and accepting them without evaluation encourages curiosity and fosters the confidence students need to examine images and representations throughout the entire lesson. According to the National Research Council (2001), "Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas." Students need to experience varied representations regularly and connect them to mathematical ideas in order to solidify their understanding.

A teacher can use the launch to pique students' interest in the upcoming math topic and explore new vocabulary as it comes up naturally in discourse. Annotating allows students to see mathematics become visible as a classmate speaks. Ideas transform into images and representations, and they become accessible knowledge for all members of the classroom. "Visual representations are of particular importance in the mathematics classroom, helping students to advance their understanding of mathematical concepts and procedures, make sense of problems, and engage in mathematical discourse." (Arcavi 2003; Stylianou and Silver 2004). Additionally, "Math drawings and other visual supports are of particular importance for English language learners, learners with special needs, or struggling learners, because they allow more students to participate meaningfully in the mathematical discourse in the classroom." (Fuson and Murata 2007)

For these reasons, I suggest teachers begin class with one image that all students can see comfortably. Looking at a common image strengthens the classroom community, whereas looking at images independently can cause students to lose focus.

The IM curriculum uses warm-ups to launch lessons and invite students into the mathematics ( More about IM warm-up routines). In each of these routines, the teacher notes succinctly state:

Display the image.

Give one minute of quiet think time.

Share and record responses.

These short, bulleted notes are so much more than teacher directions. The practice of annotating a mathematical image, equation, or expression as students share their ideas seems like a simple act, but there is much more going on than meets the eye. Principles to Actions, an influential mathematics publication, notes that it is imperative to "elicit, value, and celebrate varied approaches and solution paths..." while annotating and acknowledging student ideas. It describes the importance of developing "...environments in which students feel safe to engage with one another and with teachers." To create this kind of rich, productive community, teachers must be intentional with their annotation process.

Some examples of the decisions to make:

What will I write down?
Be selective and choose the right words to capture in your annotations. The example below shows how a student may respond to a question comparing corn and broccoli in a bar graph. The purple teacher notes focus on the important mathematical aspects of the student's reply. "Teachers must determine how to build on and honor student thinking while ensuring that the mathematical ideas at the heart of the lesson remain prominent in class discussions." (Engle and Contant 2002)

What should I do when a student shares an inaccurate idea?
Research tells us that "...it is critical not to evaluate their statements so you can hear from as many students as possible. Many students will not risk sharing their statements if they feel they will immediately be critiqued." (Seda and Brown 2021) This means that even if a student says something incorrect, criticism should not be our first response. Instead, it can be an opportunity for deeper exploration. For example, a student saying "All rectangles are squares,"' when the inverse is true can be a tricky situation to navigate. We want the student to feel valued even if the response is inaccurate, yet we must provide accurate mathematical information. An incorrect comment might be navigated by writing down the student's statement and eliciting different ideas from other students. Classroom discourse involves disagreement at times. Students in a strong math community are taught to respectfully disagree and think together to come to consensus. We can anticipate that another student will say the correct converse, "All squares are rectangles." The teacher has an opportunity to talk about how the characteristics of squares and rectangles can help us figure out which statement is true. Student thinking can be solicited, and a brief discussion of the shapes' features can lead to the conclusion that the latter statement is true. The teacher can wrap this up by circling or highlighting the correct statement. Then, she might mention that math ideas can be tough to decipher at times, but an organized consideration of thoughts can help us get the shapes figured out correctly. This modeling will help students become analytical thinkers. "When teachers allow students to share their thinking with others, it contributes to students' confidence in their own ability to reason." (Seda and Brown 2021)

Where will we experience the warm-up?
Options include at desks, standing in a math huddle, or sitting on the carpet. It is important to consider the ages of your students, the time of day, and the classroom configuration when making this decision. Try out various ideas and take note of what works best for your students.

How will I know students are ready to share?
Hand raising has traditionally been a classroom signal, yet research shows it often makes reluctant students cease thinking. In an IM math community, a quiet signal is best to indicate that you have an idea to share. Some students may give a thumbs-up or put up an additional finger or two to indicate they have more than one comment to make.

What sort of markers do I need?
Bold, thick markers are encouraged to annotate as students talk. The use of a thick marker means that the writing is visible to all. I suggest using color freely. It may also help to change colors with each speaker or to use similar colors for similar ideas. An example of the impact of using color is shown in the sample below:

What should I do if many students have the same thoughts or answer or reply?
Using an underlining method can show that multiple students have the same thought. If ten students think the value of an expression is 7 and two students think it is 8 my annotations might look something like this:
The repeated lines under the proposed answers acknowledge every student in the class.

Should I put student names with their thoughts?
As a general practice, writing student names with ideas is not recommended as it is time-consuming and could be distracting. However, there are times when it may be the best thing to do. If a student thinks of a way to solve a problem that is novel to the group, calling that method "Jarod's Way" might help students remember the method, and it will undoubtedly make Jarod feel proud of his thinking. Knowing when a student needs a boost and using their contribution as an opportunity to put their name next to their thinking is a small teacher move that can go a long way toward encouraging student participation.

Where should I annotate?
Options include chart paper or on a white board. Large chart paper is best to create an anchor chart that students can refer back to during the lesson or throughout a unit. A whiteboard will be best if the notes will not be needed for reference later.

Conclusion

The start of math class is an important time for building students' confidence and priming them for learning. The warm-up that kicks off a lesson is an opportunity to set the tone, engage all, and instill pride and ownership among students. It's not a two-way conversation. Instead, it's a whole-class learning experience where we want all students to feel like a valuable member of the collective discussion. This is important because "...acknowledging student contributions, and attending to culture and language play substantial roles in equalizing mathematics gains between poor and non-poor students." (Battey 2013; Cross et al. 2012; Krisker et al. 2012; Robinson 2013)

Annotation shows our students that we care about their ideas and that we want to know what they think. It also shows that we care about them as people. In a vibrant, thinking math learning community all contributions are valued. By writing at least one aspect of a student's thinking, we show that we respect their minds and feelings. When students feel heard, they feel good about themselves. When we listen and annotate, the goal of the launch has been reached. Students are interested, confident, and ready to learn.

Next Steps

After a lesson, consider the following to reflect and grow:

Did I acknowledge all the contributions made to the launch today?

Could I have used color in a different way?

What would I change in the future if I were to launch this lesson again?

Who didn't contribute that I might seek out and encourage tomorrow?

Jen Hawkins
IM Certified® Facilitator and Independent Curriculum Implementation Specialist

Jen Hawkins is a math consultant and facilitator for Illustrative Mathematics. She has a BA in elementary education with a concentration in mathematics and an MA from Mount Holyoke College in teaching mathematics. She has worked as a teacher and as a math coach in schools in both the Caribbean and Massachusetts. For a number of years, she left the classroom to focus on assessment and worked on the Massachusetts Mathematics MCAS test. She believes mathematics is the language of quantity and shape that everyone can appreciate and understand with the right resources and support.

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Annotate and Acknowledge - IM CERTIFIED® BLOG

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By Jen Hawkins, IM Certified® Facilitator and Independent Curriculum Implementation Specialist As I sat at the table in the back of the room, I watched the teacher reveal an image connected to the lesson's warm-up. She told her students they would have a minute to consider the image, and then she would ask them about [...]

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By Mike Henderson
As a former grade 3 teacher, I know first hand how daunting teaching math at this level can be. On top of developing fluency with addition and subtraction within 1,000, students need to learn about new topics like multiplication, division, and fractions-topics that may feel big and challenging. To add to the stress, grade 3 is often the first year with end-of-year, high stakes standardized testing. Pressure can build up for both teachers and their students and create a vicious cycle where teachers may look for gaps to close rather than look for the brilliance all students bring into the classroom.

When planning the story of grade 3 math, we had a goal of intentionally disrupting this cycle by:

ensuring that all students experience the "new" topics of multiplication, division, and fractions as clearly connected to what they already know

ensuring that all students are given time to develop deep conceptual understanding of multiplication and division, build fluency from this understanding, and have ample opportunities to apply their understanding in a variety of contexts throughout the year

In order to reach these goals, we wrote a math story that builds students' conceptual understanding of multiplication at the very beginning of the school year.

The invitation to grade 3 math

Much like the early units in Grades K-2, Unit 1 in grade 3 begins by inviting students and their teachers to learn more about one another by asking questions, collecting data, and organizing the data with graphs.As students learn more about each other, their interests, and the different ways they do math, they build the foundations of their math community.

The scaled picture and bar graphs also provide a natural opportunity to think about ways to group and count objects other than by counting by 1. This is an extension of students' work in grade 2 with skip-counting by 2, 5, and 10. The activities help students connect their understanding of different ways to group or count objects to build a conceptual understanding of multiplication. Rather than present multiplication as something new, the activities provide opportunities for students to see multiplication as a new way of thinking about math they've done before.

Students make sense of different ways to represent equal groups. The group sizes are intentionally 2, 5, and 10 throughout this unit to continue to build on students' strengths and allow them to focus on sense making rather than on calculation. They are introduced to different representations they may use throughout the year to represent their thinking, including equal groups drawings, tape diagrams, arrays, and expressions.

IM K-5 Math: Grade 3 Unit 1 Lesson 9 Activity 1 and Lesson 10 Activity 1
To learn more about how the work with arrays in Unit 1 builds coherence with students work from grade 2, check out Fluency Development Within and Across the Grades in IM K-5 Math™, part 3: Multiplication and Division .

In Unit 2, "Area and Multiplication," students deepen their conceptual understanding of multiplication through the study of area.Area concepts are introduced early on in the story of grade 3 because they build on the geometry, measurement, and multiplication foundations from K-2. As early as kindergarten, students learned that they can build a shape by putting together other shapes and can make new shapes by decomposing a given shape. In grade 2, students partitioned rectangles into equal-size squares. So, although defining area and measuring area are new, area concepts build on what students already know, and the activities in Unit 2 intentionally help make these connections.

Students use unit squares to measure the area of rectangles and relate the structure of gridded rectangles to the structure of arrays. They begin to find the value of products that include group sizes other than 2, 5, and 10. As they progress from gridded rectangles to rectangles with only the sides labeled, they make sense of the area diagram that will be used in later units.
6 x3 = 18
IM K-5 Math: Grade 3 Unit 2 Narrative

By the end of Unit 2, students have developed a stronger conceptual understanding of multiplication. They have engaged in sense-making about representations that they will use throughout the course. This conceptual foundation allows students to build fluency with multiplication and division throughout the year. For more, check out By the End of Grade 3: Developing Fluency with Multiplication by Grade 3 Lead Writer Zack Hill and Fluency Development Within and Across the Grades in IM K-5 Math™, part 3: Multiplication and Division by 2-5 Lead Writer Dionne Aminata and Grade 5 Lead Writer Sarah Caban.

Building on foundations
Addition and subtraction
In Unit 3, the focus shifts to "Wrapping Up Addition and Subtraction Within 1,000," as well as rounding within 1,000. The unit is placed here, rather than at the beginning of the year, for a few reasons. First, we want students and their teachers to make sense of multiplication in accessible ways as early as possible to give time to build fluency across the year. If addition and subtraction were the beginning of the story, we worry that any unfinished learning students have from K-2 may result in decisions that delay students' opportunities to engage with grade-level work.

Second, the work of Unit 3 allows students to revisit addition and subtraction with familiar base-ten representations. This focus on place value allows teachers to identify student strengths that can be used as they begin to multiply and divide with larger factors and dividends in later units, including multiples of 10. Unit 3 also revisits the number line as students think about rounding to the nearest multiple of 10 and 100 before they use the number line to make sense of fractions as numbers in Unit 5.

Last, the work of Unit 3 includes early opportunities for students to make sense of two-step word problems that include addition, subtraction, or multiplication. Students will continue to have opportunities to build their fluency with addition and subtraction throughout the year in warm-ups, centers, and their work with measurement in Unit 6.
Making sense of division
In Unit 4, students build on their conceptual understanding of multiplication to make sense of division. They connect division to multiplication with an unknown factor to make sense of and solve "how many in each group" and "how many groups" situations.

IM K-5 Math: Grade 3 Unit 4 Section A Narrative

Students also build on work with area and area diagrams to use the distributive property to decompose factors to make it easier to find the value of products.

IM K-5 Math: Grade 3 Unit 4 Section C Narrative
Fractions
Unit 5 is called "Fractions as Numbers." In grades K-2, students learned about composing and decomposing shapes from equal-size pieces. They learned names of these equal-size pieces-halves and fourths-in grade 1 and thirds in grade 2. Now, in grade 3, students learn that they are not simply names of equal-size pieces or parts of shapes, but are numbers. Students partition fraction strips and rectangles as they deepen their understanding of fractions and fraction notation. Later they use the number line to locate fractions and compare fractions with the same numerator or denominator.
Consolidating learning with four operations
In Units 6 and 7 students apply learning from previous units, and make connections within and across mathematical concepts.

In Unit 6, "Measuring Length, Time, Liquid Volume, and Weight," students connect fractions to measurement. They make connections between whole numbers and fractions on number lines and measurement tools. This builds a foundation for work with addition and subtraction of fractions and fraction equivalence in grade 4.

IM K-5 Math: Grade 3 Unit 6 Lesson 4 Activity 1

IM K-5 Math: Grade 3 Unit 6 Lesson 8 Activity 2

The unit also provides opportunities for students to continue developing fluency with addition and subtraction within 1,000 as they solve problems involving time and measurement. They have new opportunities to work with one and two-step problems that involve all four operations.

A lemonade stand at the fair makes 132 liters of lemonade a day. When Clare and Tyler stopped by the stand, there were 90 liters left.

At the end of the day, there were 56 liters of lemonade left, which the vendor put on sale in 4-liter jugs.

IM K-5 Math: Grade 3 Unit 6 Lesson 14 Activity 2

Although the overall focus of Unit 7, "Two Dimensional Shapes and Perimeter," is on geometry, students also revisit area and apply understandings about the four operations as they solve problems with area and perimeter. Unit 8, "Putting It Together," includes additional opportunities for students to practice their fluency with multiplication and division within 100.

Next Steps

From the very beginning, the story of grade 3 works to use the coherence in the standards to ensure that students do not see multiplication, division, and fractions as brand-new concepts, but as natural extensions of what they already know about math. We position topics that were a major focus in K-2-addition, subtraction, and place value-so that it supports the major work of grade 3, rather than derail it. How do these choices help give all students opportunities to access grade-level work throughout the year? How do they help students make meaningful connections to the math they already know and understand?

Check out the entire Stories of Grades K-5 blog post series:

You can also download IM's Stories of Grades K-5 free ebook!

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The Story of Grade 3 - IM CERTIFIED® BLOG

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By Mike Henderson As a former grade 3 teacher, I know first hand how daunting teaching math at this level can be. On top of developing fluency with addition and subtraction within 1,000, students need to learn about new topics like multiplication, division, and fractions-topics that may feel big and challenging.

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By Patti Drawdy and Yenche Tioanda
"Why not start the year with place value?" Kaneka Turner, Grade 4 Lead Writer, hears this question often. Isn't making sense of and operating on large numbers pretty essential in grade 4? Indeed, they are. Yet students begin Grade 4 in IM K-5 Math by playing with rectangles. In the first few lessons, they use square tiles to build rectangles of certain side lengths or areas. Why?

The Invitation to Grade 4 Math

As in all IM courses, the flow of units in Grade 4 was considered with care-with both the mathematics and the students in mind. And as is typical with an IM course, Grade 4 opens with an invitation: to play, observe, and engage with content that is both relatable and fresh, and that improves students' access to the major work of the grade. The opening unit also strives to generate conversations-around ideas, ways of thinking, and the experiences of the thinkers. These conversations serve as a basis for building a positive and inclusive learning community, an integral part of all first units IM K-5 Math.

The exploration of rectangles in the opening unit, Factors and Multiples, follows the design principles above. It activates concepts of multiplication of whole numbers and area of rectangles from grade 3. Then, it prompts students to make new observations about the relationship between pairs of side lengths and area, which paves the way for the introduction of factor pairs and multiples. The geometric context is also purposeful in that it grounds prime and composite numbers in concrete representations before students make sense of such numbers in situations that require more abstraction.

The experiences here develop students' multiplicative reasoning, allowing a wider range of learners to access upcoming work on fraction equivalence.

From Factors and Multiples to Fraction Equivalence

The work in Unit 2, Fraction Equivalence and Comparison, also builds ideas introduced in grade 3: the size of fractions, the location of fractions on a number line, and equivalent fractions. They use these insights to compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Students continue to engage in multiplicative reasoning. They explore doubling and halving-first by repeatedly folding strips of paper in half, which doubles the number of equal parts with each fold, and then by halving the parts in tape diagrams, which creates twice as many parts and smaller unit fractions. Later, they partition segments on tape diagrams and intervals on number lines into other numbers of equal parts. Such repeated reasoning allows students to more readily relate fractions in which one denominator is a multiple of another denominator and to see fraction equivalence.

Partitioning a diagram into equal parts becomes increasingly inconvenient as the denominator gets larger. It'd be easy enough to partition a number line into, say, 10 parts, but unwieldy to partition one into 100 parts. This practical limitation motivates a need for generalization, which supports students in relating tenths and hundredths. Understanding of factors and multiples also comes into play as students generalize the process of partitioning a number line to identify equivalent fractions.

Opportunities to strengthen fluency with multiplication are present not only in the core activities but also in the warm-ups, as shown in this sampling of Number Talks from the first two units. The numbers in each sequence encourage students to apply the distributive and associative properties to mentally find the value of each expression.

New Numbers, Familiar Operations

Having spent time deepening their understanding of the size of fractions, students are now better equipped to see-more abstractly than in grade 3-how fractions can be composed and decomposed, and like whole numbers, be added, subtracted, and multiplied. This work happens in Unit 3, Extending Operations to Fractions.

In grade 3, students saw multiplication in terms of equal groups of whole numbers. Here, they extend that insight to include equal groups of fractional amounts. Through repeated reasoning with diagrams and equations, students see regularity in the product of a whole number and a fraction, namely, that a fraction $\frac{a}{b}$ is $a\times \frac{1}{b}$, and that $n\times \frac{a}{b}$ gives $\frac{n\times a}{b}$.

Everyday contexts are used to help students make sense of non-unit fractions as products and sums. For instance, in the context of using only $\frac{1}{4}$-cup and $\frac{3}{4}$-cup measuring cups to measure the ingredients in a recipe, $\frac{5}{4}$ can be understood as a product (5 x $\frac{1}{4}$, or using the $\frac{1}{4}$-cup 5 times), a sum of unit fractions with the same denominator ($\frac{1}{4}$+$\frac{1}{4}$+$\frac{1}{4}$+$\frac{1}{4}$+$\frac{1}{4}$, or repeatedly adding $\frac{1}{4}$-cup), or a sum of unit and non-unit fractions ($\frac{1}{4}$+$\frac{1}{4}$+$\frac{3}{4}$, or adding $\frac{1}{4}$-cup twice and then adding one $\frac{3}{4}$-cup).

Later, number line diagrams, which in earlier grades supported students in reasoning about addition and subtraction of whole numbers, now support them in adding and subtracting fractions with the same denominator.

This representation also helps students see that a fraction greater than 1 can be decomposed into a whole number and a fraction, and then be expressed as a mixed number.

Students rely on the understandings developed in Units 1 and 2 to write equivalent fractions in tenths and hundredths, and then find sums of tenths and hundredths. For instance, to find $\frac{3}{10}$+$\frac{15}{100}$, they reason that $\frac{3}{10}$ is equivalent to $\frac{30}{100}$, so the sum is $\frac{30}{100}$+$\frac{15}{100}$, which is $\frac{45}{100}$ (IM K-5 Math: Grade 4 Unit 3 Overview ). The reasoning here paves the way for students to learn about decimal notation in the next unit.

Place Value, Purposefully Postponed

In earlier units, students began to develop a mental concept around tenths and hundredths. In Unit 4, they connect decimal fractions to numbers and place value in base ten.

Unit 4, From Hundredths to Hundred-thousands, begins with small numbers. Students learn about decimal notation, reason about the size of tenths and hundredths written as decimals, locate decimals on a number line, and compare and order them. They are prompted to notice the structure of the decimal notation and how it relates to place value.

After attending to place value in small numbers, students shift their attention to place value in larger numbers. They use base-ten blocks and diagrams to build, read, write, and represent whole numbers beyond 1,000. Working with numbers within 1,000,000 is quite a leap from the incremental increases in the numbers being explored in kindergarten through grade 3 (such as working within 20, 100, or 1,000), but grade 4 students are ready and can generalize more broadly. They observe that ten-thousands are related to thousands in the same way that thousands are related to hundreds, and hundreds are to tens, and tens are to ones-each base-ten unit is 10 times the next smaller unit.

Multiplicative reasoning and the idea of multiples comes into play as students reason about the size of large numbers. Students use their observations of structure to compare, order, and round numbers (by thinking about multiples of 10, 100, 1,000, 10,000, and 100,000 that are the closest to the number).

With a deeper understanding of place value, students are better prepared to consolidate their learning about addition and subtraction of whole numbers and to develop fluency as intended within the K-5 progression of addition and subtraction .

A Closer Look at Multiplication and Measurement

Unit 5, Multiplicative Comparison and Measurement, comes next. Since Unit 4 ends with addition and subtraction of multi-digit numbers, why not go right into multiplication and division of multi-digit numbers? This is because students would benefit from first developing deeper understandings around multiplication and division, measurement, and problem solving before venturing to learn ways to find products and quotients of larger numbers. In other words, the work in Unit 5 gives students better access to the major work of multiplying and dividing multi-digit numbers.

Up to this point, students have understood equal-size groups. Here, students make sense of multiplication as a way to compare quantities. In prior grades, they made additive comparisons, asking "how much more or less?" In multiplicative comparison, the key question is "how many times as much or as many?" Students use diagrams and equations to reason about situations that involve comparing multiplicatively.

Multiplicative comparison is also helpful in relating units of measurement. For instance, we generally think of 1 yard as 3 times as long as 1 foot, rather than as 2 feet longer than 1 foot. Solving problems that involve measurement conversion and fractional values, though classified as "supporting work" of the grade, also reinforces concepts on fractions and operations that are central in grade 4.

Upon their arrival in Unit 6, Multiplying and Dividing Multi-Digit Numbers, students are more ready for this major work of the grade. They can see the benefits of decomposing factors or dividends by place value, make sense of different representations-diagrams, equations, and algorithms-being used to organize partial products or partial quotients, and make connections between reasoning strategies and representations.

Put another way, students are better equipped to notice and make use of structure-of base-ten numbers, properties of operations, and the relationship between multiplication and division-to multiply and divide larger numbers effectively and strategically. The progression of the work in Units 1-6 helps to put students on a pathway toward fluency with these operations as expected in their K-5 journey .

Next Steps

At first glance, the story of grade 4 may seem to follow a curious path. We hope that a closer look shows a coherent sequence that allows students to make stronger connections across topics and opens the access to grade-level work for a wider range of learners. We also invite your reflections.

What other considerations may support the placement of multi-digit operations and place value work in this course?

In what other ways might the design decisions improve students' chance of reaching grade-level standards and empower them to know, use, and enjoy mathematics?

Check out the entire Stories of Grades K-5 blog post series:

You can also download IM's Stories of Grades K-5 free ebook!

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The Story of Grade 4 - IM CERTIFIED® BLOG

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By Patti Drawdy and Yenche Tioanda "Why not start the year with place value?" Kaneka Turner, Grade 4 Lead Writer, hears this question often. Isn't making sense of and operating on large numbers pretty essential in grade 4? Indeed, they are. Yet students begin Grade 4 in IM K-5 Math by playing with rectangles.

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By Mike Henderson
Teaching addition and subtraction in grade 2 is a challenging balancing act. As students move from numbers within 100 to numbers within 1,000, they need to use approaches that involve directly representing the quantities they are adding or subtracting and be supported in reasoning more abstractly. However, we don't want students to feel they always have to represent the quantities they add and subtract with objects or drawings, especially within 20. Although we want students to use increasingly abstract and efficient strategies based on place value, we don't want to push them too fast toward the algorithms they will use in grades 3 and 4 or position these strategies as the "right" way to add or subtract.

When considering different ways to tell the mathematical story of grade 2, we wondered:

How do we make sure students are given time to consolidate their understanding of numbers and operations within 100 before extending their place value understanding within 1,000?

How do we make sure students spend time making sense of place value strategies without feeling forced to use procedures that do not make sense to them?

How do we make sure students (and their teachers) do not feel pressure to master a laundry list of different strategies and representations for adding and subtracting?

We know students enter grade 2 with a variety of approaches for reasoning about adding and subtracting numbers. We want them to know that the ways they think about addition and subtraction continue to make sense even as the numbers get substantially larger. Although understanding place value and using place value understanding to add and subtract is the major work of the grade, a goal in writing the story of grade 2 was to make sure students focus on foundational math concepts without feeling funneled toward procedural ways of thinking.

The invitation to grade 2 math

The initial lessons of grade 2 invite students to learn more about each other and the ways they do math through activities and games that focus on addition and subtraction within 20. They continue to learn more about each other and their interests by asking questions and organizing data with picture graphs and bar graphs. The activities provide opportunities to build a joyful math community where students are encouraged to share ideas and celebrate different ways of thinking.

Teachers learn about the ways students think about addition and subtraction. An important focus in the early lessons is on combinations of 10 and making 10. These early activities help students not only recall their work with combinations of numbers that make 10 from kindergarten and grade 1, but also recall their work with place value. Later activities in the section give teachers an opportunity to learn about how students think about tens and ones as they represent numbers within 50 and add them.

The picture graphs and bar graphs also provide an accessible way for students to show different ways they may add or subtract. As they answer questions about the graphs, students show different ways to add and compare quantities within 20, including strategies like counting on. Although counting strategies as students' go-to method will fade throughout the year, it is a useful strategy that makes sense to many students and one we want students to continue to use with flexibility. The number range is intentionally kept within 20 in this section to make the work accessible to all students, while nudging students to reason more abstractly about the graphs.

Teachers also have opportunities to learn about how students add within 100 with a focus on how they think about adding using their understanding of place value. The numbers used are intentionally within 100 to give a purpose for introducing a more abstract representation to make sense of the quantities in Compare problems-the tape diagram. Although the tape diagram is not a representation used to make sense of performing computations, it does help students make sense of the relationships between quantities in story problems and apply what they know about the relationship between addition and subtraction to solve problems.

Subtract with strategies based on place value

Where Unit 1 elicits what students already know about addition and subtraction, Unit 2 extends their understanding of subtraction within 100 using strategies based on place value and other ways that make sense to them. Students build on counting strategies to consider how they might count on by tens and ones.

Make trains with cubes.

Find the total number of cubes you and your partner used. Show your thinking.

Find the difference between the number of cubes you and your partner used. Show your thinking.

Students are then introduced to base-ten blocks as a way to represent tens and ones. Students explore these blocks and compare them to their use of connecting cubes. Although students may continue to use connecting cubes throughout the unit, as the numbers increase in size, they begin to see the benefit of using base-ten blocks to quickly show tens and ones.

This leads to more explicit conversations about place value. Students show their understanding of the value of each digit as they explain the ways they represent situations using base-ten blocks and expressions.

Students make sense of base-ten drawings. Students first see and use these drawings in grade 1 when they are a direct representation of the connecting cube towers of 10 and single cubes. Now, students may interpret them as connecting cubes or as base-ten blocks. Over the course of the year, students continue to use base-ten blocks and drawings to represent decomposing tens in different ways.

Although the unit introduces base-ten blocks and decomposing a ten as a strategy for subtraction within 100, both are built on ways students have reasoned about and represented operations with two-digit numbers in grade 1. Students are asked to make sense of and try out different ways to represent decomposing a ten when subtracting by place, but they are not expected to use one particular strategy or representation at the end of the Unit 2 to subtract. Students should continue to use the approaches that make sense to them.

Deep dives on length and a hundred
Length and the number line
In Unit 3, students extend their understanding of measuring length as they make sense of and use standard length units. Students measure, compare, and add lengths using a variety of concrete and abstract tools. This helps all students continue to consolidate their understanding of the structure of the number system and prepares them to make sense of the number line. Students' understanding of length units and the way length units are represented on a ruler is essential to work with the number line for addition and subtraction in Unit 4 and lays a foundation for students' understanding of fractions as numbers in grade 3.

In Unit 4, students continue to deepen their number sense within 100 as they locate numbers and reason about their relative size. They also use the number line to deepen their understanding of addition and subtraction. In particular, the number line helps students think about subtraction as the distance or length between two numbers on the number line. This understanding helps students flexibly choose computation strategies within 100 and decide when some strategies might be more useful than others. For example, compare these different approaches for finding 81 - 79 that students investigate in Unit 7 Lesson 1.

Although the number line is not a representation that easily shows composing or decomposing base-ten units, it does help students show and explain strategies based on counting on or counting back by place. Students also see the strategies they used when adding and subtracting within 20-such as making or getting to a multiple of 10-continue to work with larger numbers.

IM K-5 Math: Grade 2 Unit 4 Lesson 10 Activity 2 and Unit 4 Lesson 11 Activity 1
A hundred
In Unit 5 students make sense of a hundred as a base-ten unit.

Finally! Three-digit numbers and place value! Many teachers may recognize the content of this unit as the traditional focus of early units of grade 2. Although we know that many students are ready to extend their understanding of place value to include a hundred at the beginning of grade 2, we also believe that in building on students' understanding and strengths with working within 100 in earlier units, all students will be able to make sense of a hundred as a unit and will be less likely to feel like strategies based on place value are the only (or "right") ways to add and subtract.

Because the introduction of three-digit numbers comes after extensive work with numbers on the number line, students are also able to use this tool as a way to compare three-digit numbers. Although the focus in the unit is on using place value reasoning to compare numbers, the number line supports students' ongoing development of number sense and counting within 1,000. The development of a strong sense of the relative size of three-digit numbers helps later when students consider the relationship between numbers in an expression before selecting the approach for finding its value.

How do both representations show that 432 > 423? How does each representation connect to students' development of strategies for addition and subtraction within 1,000?

Consolidation

In Units 6, 7, and 8, students consolidate their understanding about addition and subtraction within 100 and their understanding of place value within 1,000. In Unit 6, students extend their understanding of number lines and skip-counting to tell time to the nearest hour and minute on analog clocks. They also add and subtract within 100 as they solve problems with money. In Unit 8, students skip-count and use expressions with equal addends to determine whether quantities are even or odd, and to find the total number of objects in an array. This is the foundation for their work with multiplication in grade 3.

In Unit 7, students add and subtract within 1,000. No new strategies or representations are introduced in this unit. Students build on the ways they have added and subtracted numbers since Unit 1 with the ways they represented numbers within 1,000 in Unit 4. This includes inviting students to think about ways they can use the relationship between addition and subtraction and counting on strategies.

They make sense of ways to compose and decompose units, and explore different ways to represent their thinking. No one approach is positioned as the "right" way to add or subtract by place, and the emphasis is intentionally on place value concepts rather than procedures to avoid moving too fast toward the standard algorithm.

The unit lays a foundation for understanding and using algorithms in later grades. For example, the way we present drawings to students in the materials mirrors the annotation used in many subtraction algorithms.

Mai used base-ten blocks to find the value of 336 - 52. Then, she started making a diagram to show her work.

Explain what Mai did in Step 2. Show what Mai could do next to find the difference.

IM K-5 Math: Grade 2 Unit 7 Lesson 16 Activity 1

Students also see recording methods that are precursors to the algorithms they will see in grade 3 and build off the ways they represented three-digit numbers in Unit 4. However, they are not expected to use or master these recording methods. The mathematical purpose is for students to make sense of these methods and to invite students to try them. They connect the ways that make sense to them to these more abstract recording methods. They also have opportunities throughout the unit to share their own thinking, including any invented algorithms, and reflect on ways to make their thinking clear for themselves and others.

Throughout each section of the unit, students flexibly choose strategies and explain why they chose a particular strategy based on the numbers in the expressions. They continue to see that the strategies they learned in earlier grades continue to work with larger numbers.

Jada's thinking for 402 - 298

What ways might you expect students to find the value of these differences? How might they show that they are thinking about the relationship between the numbers and choosing strategies flexibly and strategically? How do these strategies connect to the strategies they shared as early as Unit 1?

Next Steps

From the beginning of grade 2, students use the approaches that make sense to them as they add and subtract within 100. They are introduced to a limited selection of new representations-base-ten blocks and number lines-as ways to represent their operational thinking. Each strategy they use is an extension of strategies they have used before with smaller numbers. How do these choices in our story help students develop an understanding that math makes sense? How does it prepare students for the work of adding and subtracting with algorithms in grade 3 and the standard algorithm in grade 4?

Check out the entire Stories of Grades K-5 blog post series:

You can also download IM's Stories of Grades K-5 free ebook!

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The Story of Grade 2 - IM CERTIFIED® BLOG

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By Mike Henderson Teaching addition and subtraction in grade 2 is a challenging balancing act. As students move from numbers within 100 to numbers within 1,000, they need to use approaches that involve directly representing the quantities they are adding or subtracting and be supported in reasoning more abstractly.

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The Story of Grade 1 - IM CERTIFIED® BLOG

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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 marginalized students-often students of color-between the grade-level expectations laid out in standards and students' opportunities to engage with this content in their math classes.

Oftentimes, when students are perceived as being not "ready" for content, they are often limited to below-grade-level work. This practice disproportionately occurs in classrooms where the majority of students are students of color.

from The Opportunity Myth (2018)

If we want to close the opportunity gap, we have to do many things differently than we have done in the past, one of which is re-examine the sequence of the content we put in front of students. The IM K-5 Mathematical story of grade 5 was intentionally written so all students have access to grade-level content.

Many grade 5 math curricula start the year with units on place value concepts and whole number multiplication and division algorithms. When considering whether to start the year this way, we wondered:

What message does the mathematical content in the early units of the year send to students about what it means to "do math" and what is valued in math class?

What assumptions might educators make about students based on their successes or challenges with multiplication and division algorithms at the beginning of the year?

How might these assumptions impact students' access to grade-level content throughout the course of the year?

Extend an Invitation

From the start of the year, we want to send the message that mathematics is an opportunity to be curious, collaborative, and playful. To do this, we begin with the concept of volume. Because this concept is introduced for the first time in grade 5, students do not have any previous negative experiences or connotations. Students are invited to be curious and creative, and teachers have opportunities to notice and build on what students do know and can do. It also diminishes, if not eliminates, the assumptions both teachers and students might make about what it looks like to be "good at math."

In early lessons, students have many opportunities to build with unit cubes and use mathematically informal words and phrases to explain the layered structure of rectangular prisms. Teachers listen for and display the words students use-such as "slices," "groups," or "layers,"-to describe how they counted the cubes to determine the volume. Students connect their less formal language to multiplication expressions that represent the volume of rectangular prisms before they are introduced to more formal math vocabulary and procedures.

How do the expressions 5×24 and 6×20 represent the volume of the rectangular prism? Explain or show your reasoning.
Over time, students connect their informal strategies for finding the volume of rectangular prisms to generalized formulas. While students are developing conceptual understanding of volume, the quantities used in the problems were deliberately chosen to be "friendlier" numbers. This allows students to strengthen their fluency with multiplication in advance of the work on the standard algorithm for multiplication, which is introduced in Grade 5, Unit 4. After studying the standard algorithm, students revisit the concept of volume, solving problems that involve larger multi-digit numbers as side length measures. This progression allows students time to develop big, conceptual ideas and reinforce fluency work from earlier grades before demanding procedural fluency in new grade 5 content. It also decreases the likelihood that students will be perceived as not ready for new grade-level topics because of their proficiency with computation. We believe these choices will increase student access to grade-level content.

Put Fractions at the Forefront

Throughout the year, we want to maximize students' opportunities to access grade-level content. In Units 2 and 3, we introduce grade-level content while providing ongoing opportunities to reinforce prior grade-level understandings and fluencies, all within the work of fractions. While many curricula organize whole number computation before fractions, this sequence has implications for extending an opportunity gap. If students are perceived as not being able to use whole number multiplication and division algorithms correctly, they may be confined to remediation cycles and denied the opportunity to engage with grade-level fraction concepts and procedures. We know that understanding fraction operations is a major indicator of success with topics in high school mathematics so we want to ensure that each student has the opportunity to make sense of grade-level fraction understandings and operations early in the year ( Siegler et. al., 2012, p. 695).

Continue to Build on Prior Grade-level Content

In Units 2 and 3, students revisit the meaning of fractions, multiplication, and division. They connect what they know about these topics to new learning.

Students begin Unit 2 by interpreting a fraction as division of the numerator by the denominator. Through this work, they continually deepen their conceptual understanding of fractions and division. In the example below, students connect what they know about unit fractions and division.

As the unit progresses, students extend their understanding of multiplication as they multiply a whole number by a fraction. In the example below, students leverage what they know about whole number multiplication to solve problems involving multiplication of a whole number and a unit fraction.

Find the area of the shaded region. Explain or show your reasoning.

In Unit 3, students continue to use area diagrams to make sense of fraction multiplication. They build on the new understanding of multiplying a whole number by a fraction while applying prior-grade understanding of area. They recognize that, when multiplying fractions, the unit is the size of the piece being tiled within the unit square. They count those unit tiles by multiplying the number of tiles in each shaded row by the number of shaded rows.

For example, when asked to determine the area of the shaded region below, students know that the unit square is partitioned into a $4 \times 5$ array, so the size of each tile is $\frac{1}{20}$. They then multiply $2 \times 3$ to find how many tiles are shaded and determine that the area of the shaded region is $\frac{6}{20}$.

Students later apply these understandings to multiply any two fractions.

Students are offered another opportunity to revisit the major work of prior grades in the context of the major work of grade 5 when they are introduced to division of a unit fraction by a whole number and a whole number by a unit fraction. As shown in the example below, the sequence we chose offers students the opportunity to be curious and creative while deepening and extending their understanding of the meaning of division.

Sequencing fraction multiplication and division before whole number algorithms invites teachers to notice and celebrate what students do know about multiplication and division, and minimizes the chances teachers and students will make assumptions about what students don't know about fraction operations based on their multi-digit computation.

Make Algorithms Accessible

By the time students encounter whole number multiplication and division algorithms in Unit 4, they have had significant experience building the necessary fluencies and understandings to be successful. Teachers have had significant opportunities to observe what each student knows and how they apply these understandings.

In telling the story of grade 5, we chose to begin with an invitation to engage in grade-level mathematics, introduce fractions early in the year, and postpone multi-digit multiplication and division algorithms until later in the year. We think these choices will:

send the message that each student is creative, curious, and capable of doing grade-level math in the beginning of the year.

support teachers to notice and build on what students know and can do.

increase opportunities for students to access grade-level content.

We hope these choices lead to a world where more historically marginalized students have the opportunity to know, use, and enjoy mathematics.

Next Steps

IM's Grade 5, Unit 1 is available at the IM Community Hub.
Try it out! Tell us how you and your students found opportunities to be curious, collaborative, and playful.

For more about how the story begins for other grades in IM K-5 Math, read Kristin Gray's blog post, " First Impressions: The First Units in IM K-5 Math."

Check out the entire Stories of Grades K-5 blog post series:

You can also download IM's Stories of Grades K-5 free ebook!

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The Story of Grade 5 - IM CERTIFIED® BLOG

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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 marginalized students-often students of color-between the grade-level expectations laid out in standards and students' opportunities to engage with [...]

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The Story of Kindergarten - IM CERTIFIED® BLOG

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By Alexandra Clayton Many 5 year olds love to tell you (and show you) how high they can count. This skill of rote counting, saying the counting words in the same order each time, is a lot like singing the ABCs . While students may sing "l-m-n-o-p", they may not know that "m" is a [...]

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Explore how to successfully implement centers into your instruction.