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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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Imagine Learning Illustrative Mathematics Success Story

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Imagine Learning Illustrative Mathematics Success Story

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Watch this short video to see how IM with Imagine Learning is positively effecting both students and teachers.

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Imagine Learning Illustrative Mathematics Success Story

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We are excited to help in your journey of reviewing Imagine IM! Please use this padlet to access valuable information of various aspects of the program. As you move left to right, some sections will have vertical scrolling capability, where you can find even more valuable information.

Should you have any questions, please reach out to your account executive:

Marla Bormann (marla.bormann@imaginelearning.com)

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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. (available in grades 6-9)

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URL: https://imagineim.ilclassroom.com/

Username: bellevillepsd118@example.com

Password: teacher

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Building Thinking Classrooms with Imagine IM

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Building Thinking Classrooms with Imagine IM

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Building Thinking Classrooms with the IM Curriculum - IM CERTIFIED® BLOG

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By Adrienne Baytops Paul, Math Specialist, UnboundEd, COO of My Mathematical Mind, and former Middle School Teacher The 14 Practices of Peter Liljedahl’s Building Thinking Classrooms (BTC) complement the Illustrative Mathematics curriculum in remarkable ways. In my 23 years as a classroom teacher, I…

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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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Significant Growth in Student Understanding

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Significant Growth in Student Understanding

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Imagine IM: Enhanced Illustrative Mathematics

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This curriculum is a 2-year version of the IM 6-8 Math curriculum, allowing students who are ready to begin high school mathematics in Grade 8.

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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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In a class of 25 students, there are 25 different learners with 25 different backgrounds. Students come into the classroom with varied experiences with school, with mathematical content, and with life. While this has been true for time immemorial, teachers may feel it more acutely since the onset of the pandemic. The IM Team would like to support teachers with planning for the variety of student entry points. In particular, what features in the curriculum support strategies for providing additional time on prior content?

IM's Approach to the Diversity of Student Entry Points

IM recognizes that students approach new mathematical learning with different entry points. Some teachers may have experienced a wider variety of entry points due to disruptions to the 2019-2020 and 2020-2021 school year. Other teachers may consistently experience a wide range of entry points, for reasons including disruption in formal education, lack of full access to content in prior grades, or because some students take more time to fully know, use, and enjoy content.

Extensive remediation on below grade-level content is not productive. When students have difficulties accessing grade-level work, the approach of giving students work from earlier grades is ineffective, and approaches that provide access to grade-level work are preferred. A recent study by TNTP showed that giving students work from earlier grades results in losing more academic ground and getting even less access to grade-level work in the future. In other words, "meeting students where they are" and trying to remediate learning deficits often results in having to meet them even further back next year. Those stuck in this cycle are disproportionately black and brown students, students experiencing poverty, students with learning disabilities, and students learning English.

Learning new ways to support students coming in with different entry points is challenging, but ". . . trying to give every child a real chance to do grade-level work, however imperfectly, will lead to far better results than picking and choosing who gets those opportunities-right now and over the long run." ( Steiner and Weisberg, 2020 )

IM is designed to provide access and challenge to all students within the entire, inclusive classroom. All students benefit from being part of the conversations and hearing multiple perspectives and approaches from their peers, and the curriculum is designed with a wide variety of entry points in mind.

Teachers take advantage of built-in features of the curriculum to respond to student thinking during grade-level instruction, including:

accessible tasks designed to invite a variety of responses and give students a chance to learn from studying multiple approaches

warm-up activities and activity launches designed to provide an invitation to the mathematics and activate prior knowledge in the service of accessing grade-level content

prompts in each activity to address misconceptions

IM has several built-in features to position each teacher to respond to student thinking in real time. There are suggested adjustments based on student responses to cool-downs and pre-unit diagnostic assessment questions. There are also additional resources, such as the Algebra 1 Extra Support materials. When a teacher is responding to student thinking during the lesson, each student's learning is being addressed just in time and not postponed until later with a separate experience.

Strategies for Providing Additional Time on Prior Content
Given the wide variety of starting points students come in with, teachers may find that some students need additional time and additional instruction to be ready for grade-level content.
Schools provide additional time and additional instruction in a variety of ways, depending on student needs and available resources.
Support for the whole class
When most students in a class need some additional instruction to access grade-level content, teachers might use an optional lesson already in the curriculum, or use the suggestions in a Curriculum Adaptation Pack to find an activity, lesson, or practice problem from a previous grade or unit to supplement.

Adding supplemental material from a prior grade so that students can access grade-level content should be:

Just in time, rather than preventative. Rather than beginning the year with a long diagnostic assessment and spending several weeks covering prior-year material, use the Check Your Readiness pre-unit diagnostic questions and Curriculum Adaptation Packs to determine interventions needed just before encountering grade-level content within the unit.

Limited to what students need to access the grade-level content, so that students have a chance to do the major work of the grade. For example, in Grade 8 Unit 2, students will recognize when two slopes are equal. An unproductive approach would be to stretch back to grade 4 work on fractions of a pizza, or fractions of a dollar, or other contexts focused on visualizing equivalent fractions-this is unhelpful for the purposes of comparing slopes. Instead, the work students have just been doing in Unit 2 on similar triangles will help them in Unit 3 recognize that, for example, the slope 4/3 and the slope 8/6 are equal, and they can use different arguments for explaining why. A quick refresher on equal quotients, such as extending the warm up from Grade 8 Unit 2 Lesson 10 to include some more examples and a discussion of different strategies, some references posted in the room, and having a calculator handy may be all that is needed.

Balanced with thoughtful prioritization. IM used the guidance from Student Achievement Partners to prioritize major work of the grade in recommending lessons and topics to cut if most students need significant additional time to be ready for the grade. This ensures that students get to experience instruction on major work of the grade and don't fall further behind.

Support for individual students

Unfinished learning is not the same as special education or having an Individualized Education Plan. Sometimes these two are conflated, but many students who have unfinished learning do not have learning disabilities, and many students with learning disabilities are ready for grade-level content provided there are multiple means of access and engagement. IM's support for students with disabilities is designed to provide those multiple means per the Universal Design for Learning principles so each student has access to the grade-level content in the curriculum.

Because of this, students may need additional time focused on math for multiple reasons:

A student with an IEP may have the prerequisite understanding and skills, but may need more time to process the grade-level content being worked on or more ways to explore the content.

A student may be able, generally, to learn in the amount of time and structure of their math class but may need to understand additional content or skills from prior grades in order to be successful with upcoming content.

A student may need to learn content that they need to access upcoming content and may need more time and more means of access and engagement, representation, and action and expression to learn that content and the grade-level content being taught in their math class.

Therefore, it is important to be clear in planning whether additional time outside of math class is designed to make sure students are ready with prerequisite knowledge, or to give students who are ready for grade-level content but need more time and a wider variety of means of accessing the content, or an intervention class is serving both needs at once.

The IM curriculum does not currently have an aligned intervention product. However, when there is time outside the math block for additional support for individual students, we recommend fortifying the representations and routines from IM rather than learning other representations and routines. Being introduced to too many strategies, programs, and representations causes cognitive overload and dilutes students' ability to focus on the math content. Extra time with what a student is seeing in math class, and with just-in-time prerequisite content, gives opportunities to be more successful in math class. For the Algebra 1 course, there is a set of support materials designed for this purpose. Read more about the Algebra 1 Extra Support materials.

In order to facilitate providing additional support outside of the math block with aligned routines, representations, approaches, etc., IM has resources that schools and districts can take advantage of:

pre-unit diagnostic problems (included in IM 6-12 Math as the Check Your Readiness assessment). These can aid in determining which students may need additional time focused on prerequisite content as well as identifying the most important prerequisite content

optional lessons and activities that often provide extra practice or review prior grade-level materials

notes for teachers in each cool-down with suggestions for supporting students including suggestions for reviewing prior learning to give more students access to the grade-level content. These can help teachers identify opportunities for both supporting grade-level content and supplementing with work on prerequisite content that may be impacting students' current work

Curriculum Adaptation Packs, available on the LearnZillion platform and in the IM Community Hub, which suggest prior lessons, activities, and sections to revisit to help students get ready for upcoming work in a "just in time" fashion

Next Steps

Reflect: how do you support various student access points during math class? What structures can you put in place outside of class to support student access during class, and what might these additional math blocks look like? What parts of the IM curriculum would you want to use or reinforce?

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What Does IM Think About Unfinished Learning and Supporting Students with a Variety of Entry Points? (Grades 6-12) - IM CERTIFIED® BLOG

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In a class of 25 students, there are 25 different learners with 25 different backgrounds. Students come into the classroom with varied experiences with school, with mathematical content, and with life. While this has been true for time immemorial, teachers may feel it more acutely since the onset of the pandemic.

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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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Our IM K-12 comprehensive program is classroom ready and

enhanced with exclusive features to make instruction come alive.

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Select classroom activities include an opportunity for differentiation for students ready for more of a challenge. We think of them as the “mathematical dessert” to follow the “mathematical entrée” of a classroom activity.

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By Max Ray-Riek, Director of 6-12 Professional Learning

In my role at IM, working with teachers and administrators, I am asked to help with the challenges of implementing an IM curriculum. We are often asked, "How can I differentiate instruction while using this curriculum?" or "What resources does IM offer to support differentiation?"

When we created the teaching guide for the curriculum, we chose not to use the word differentiation . This was deliberate, as differentiation is interpreted in many ways. One common interpretation is providing different instructional tasks for different students, either through differentiated student materials or by pulling small groups based on some diagnosis and providing different instruction to those groups.

We focus on and . IM K-12 Math is designed to give all students access to grade-level mathematics, by providing tasks that students can make sense of, reason about, collaborate to solve, and learn from without specifying the strategies students must use to solve the problem. The curriculum is also designed to give all students opportunities to go deeper with the content, and to make many connections between concepts. In fact, the curriculum is designed for and works best when students come up with a range of approaches (concrete, abstract, correct, incorrect, algorithmic, visual, etc.) and are given multiple opportunities to compare their thinking to peers and to the examples provided in the curriculum. All students benefit from hearing a range of ideas and connecting their ideas to the ideas of peers with different approaches.

We recognize that students come to us with a wide variety of interests, strengths, experiences with math and language, cognitive resources, etc. Providing access to grade-level mathematics does require that teachers make strategic adjustments in order for students to engage, make sense of the questions being asked, persist, collaborate with peers, share their thinking, listen to and understand each others' thinking, internalize and try on new ideas, and practice and apply what they are learning.

Educator and researcher Rhonda Bondie suggests that teachers plan for and use differentiated instruction before, during, and after a lesson as they

adjust common instruction: adjust the launch, directions, sequence of activities

use specific resources: offer a support such as sentence frames to students who need access for participating in conversation with their classmates

individualize practice: offer various practice assignments for students to choose from (2019)

What does this look like in a problem-based classroom? When we apply Bondie's suggestions, we consider what supports are already included in the curriculum, as well as how teachers can plan to support access and challenge for all students. Let's consider an example drawn from experiences with several teachers and coaches using IM across the country. The example is from IM Algebra 1 Unit 4: Functions, and we'll call the teacher Ms. Goodson.

Adjust common instruction: Use a familiar context

Ms. Goodson is preparing to teach Lesson 1: Describing and Graphing Situations . She knows that the purpose of Lesson 1 in each unit is to invite students to the mathematics of the unit and that students will have opportunities throughout the first several lessons to continue to work on these ideas. The lesson goals will help her decide what is most important for all students to consider. She reads the lesson narrative and the learning goals. The learning goals for the lesson are:

Interpret descriptions and graphs of functions in context.

Understand that a relationship between two quantities is a function if there is only one possible output for each input.

Use words and graphs to represent relationships that are functions, including identifying the independent and dependent variables.

Ms. Goodson next looks at the cool-down and student lesson summary. This gives her a sense of what she should be monitoring for as students work-she should listen to students talking about whether one variable is a function of the other, considering whether there is only one output for a given input, and either drawing graphs from descriptions of situations or describing situations based on graphs.

With that information in mind, she takes a look at the warm-up. It's about buying bagels from a bagel shop. There is not a bagel shop in her town, though many students have seen them for sale in prepackaged bags at the grocery store. Ms. Goodson considers whether this is a good opportunity for students having a window into another culture, but is concerned about the complex context and complex mathematics being too big of a barrier.

This problem features a non-linear pricing scheme, which reminds Ms. Goodson of heated conversations students had during this year's homecoming festivities where couples and groups got discounts for dance tickets. She also remembers that at the homecoming fair tickets for the carnival games were sold individually, by the half-dozen, and the dozen. She decides that changing the context from bagel pricing to ticket pricing will help her students quickly access the warm-up and maintain the mathematical intent: surfacing the idea of the output for a given input being ambiguous (for example 13 bagels could be sold for \$11.25, \$13.00, \$13.25, or \$16.25 unless the store owner always sells other quantities of bagels at the best price).

Use specific resources: Blank copies of the table

Based on previous lessons that involved completing tables of values, Ms. Goodson anticipates that some of her students might have a hard time getting started on the task of filling out the best price table. Some students may want to argue for different pricing schemes, others may feel overwhelmed by having to think about all the different possible prices for each number of tickets and narrow down to the best price. She doesn't want students to get bogged down in calculating; she wants them to see and understand the tables and graphs that represent the ambiguous situation versus the situation where price is a (non-linear) function of number of tickets purchased. Therefore she decides to print blank copies of the table of each person's pricing scheme from the activity synthesis to have on hand as needed for groups that are struggling to get started on the best price table.

Her goal is for all students to be ready after 7 minutes of group work time to participate in a conversation about why price is not a function of number of tickets purchased, but best price is.

Use specific resources: Manipulatives

Ms. Goodson also has an idea about Activity 2: Be Right Back! While it's unusual for dogs to be tied up outside of stores in her community, it's pretty common for a dog to be tethered in a yard. However, some of the emerging bilingual students in her class might still have a hard time understanding the descriptions of how the dog moves around the post. How could they engage in a conversation about what the graph should look like if they're having trouble visualizing or asking questions about the situation?

Ms. Goodson reads the suggested supports for English learners and students with disabilities, and plans to launch the task as written in the curriculum, demonstrating the dogs' action on Day 3 using a stuffed dog leashed to a spool of thread and inviting multiple students to use the toy dog to show how they interpret the story, and then share how they might sketch a graph of the dog's distance from the pole over time. She makes sure that she has enough toy animals and spools of thread to offer them to small groups as needed to support their sense-making and conversations. She is confident all groups will be able to get started after that launch, but that they will still have to work hard to both translate the words into actions and the actions into a sensible graph.

Individualize practice

As Ms. Goodson studies Activity 1.3: Talk About a Function, she realizes that there are multiple opportunities to individualize practice and give students different amounts of scaffolding to continue to move towards the learning goals. The task itself gives students the opportunity to make a choice about which pair of variables they choose, and she realizes that some pairs of variables might be harder than others. She also thinks about ways she could provide more access, challenge, and choice for students.

Assign some students an activity from the aligned Algebra 1 Supports lesson which gets to the same learning goals but involves matching rather than creating your own graph.

Assign students a particular pair of variables to focus on first, one that she knows will be accessible and will help them get started with confidence.

Create a worksheet for students to help them remember to answer each of the questions and include sentence stems to help students explain their ideas, not just write their answers.

Encourage students to brainstorm additional pairs of variables, and choose one pair to focus on.

Give students the challenge of making up their own story for a partner to graph.

Use a website such as graphingstories.com to find more examples, based on videos, which might give students more access than a written description. She notes that in the graphingstories.com handout time is already on the -axis. Since she would like students to make a choice about which variable to put where, and how to scale the axes, she would need to provide blank graphs.

Given what she sees in Activity 1.3, and what she knows about the cool-down and lesson summary, Ms. Goodson decides that while students are doing more individualized practice during Activity 1.3, she will monitor and note students who might struggle with the cool-down, which is very similar to the practice. She knows that students will have more opportunities in the next lesson to show what they know, so she decides to invite a few students to come to her desk to work on the cool-down collaboratively. Then they spend time reading and highlighting key ideas in the student lesson summary, so she can better understand what makes sense to them now and what they are still grappling with. She will continue to monitor these students' understanding throughout the next several lessons since the cool-down guidance states that students will have more opportunities to work on these mathematical ideas. There is no need to "press pause" until Lessons 4 or 5 if students are not yet able to evaluate and interpret equations given in function notation.

Whew! That is a lot of decisions to make, but Ms. Goodson has been watching and listening to her students and uses many of the suggestions and supports already built in the curriculum and the lesson plan. Ms. Goodson spends her planning time thinking about how she will make adjustments to support her students, rather than having to create lots of additional materials or come up with different standards-aligned activities for different students.

Next Steps

Consider an upcoming lesson.

1. What supports for access and challenge are already included in the lesson?

2. What can you ask, amplify, or adjust during common instruction to provide additional access or challenge?

3. What specific resources can you offer students who need additional access or challenge?

4. How can you individualize practice using:

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Differentiating Instruction with IM 6-12 Math™ - IM CERTIFIED® BLOG

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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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Raise your hand if you have been perplexed by professional learning that told you your class needs to be culturally responsive, but left you completely unsure about what that means in a math class. Yup, been there, done that! Often the techniques, tools, and suggestions are aimed at courses where, traditionally, students write many papers and have robust discussions about the world-the humanities. I, like many other math teachers across the country, have found that this may stem from a misconception about what culturally responsive teaching is. There are universal methods for integrating its tenets into our teaching practices. Educators teaching with IM K-12 Math are fortunate because IM's language and instructional routines support culturally responsive pedagogy. This can open a door of opportunity for us to prepare students for active citizenship-to be able to critically analyze society.

As described by Zaretta Hammond, culturally responsive teaching is about "rebuilding trust with [students] through a learning partnership, and using that rapport and trust to get permission from students to push them into their zone of proximal development." For too long we've had culturally responsive teaching simplified to being about employing strategies for engaging students or solely building race-based connections. Although elements of this may exist in a culturally responsive learning space, they are topical and alone are not culturally responsive. Gloria Ladson-Billings, an American pedagogical theorist and teacher educator has identified three elements that must be true in culturally responsive learning spaces (1) students must experience academic success, (2) students must develop and maintain cultural competence, and (3) students must develop a critical consciousness through which they challenge the status quo of the current social order. (But That's Just Good Teaching!)

IM K-12 Math has several features that support culturally responsive pedagogy, including focused and coherent courses of study, attention to the selection of real world tasks and contexts, and the way lessons and activities are structured. In this blog post, I am going to focus on the way that instructional routines and activities are structured. We can use our students' cultural learning styles and tools to leverage the brain's memory systems and information processing structures. Three strategies that Hammond suggests accomplish this are "gamifying, make it social, and storifying it" and I think we'd all be pleasantly surprised to know that examples of this exist in Illustrative Mathematics' instructional and language routines.

When an assignment is "gamified," students use repetition, solve a puzzle, or make connections between things that don't seem to be related. Specifically we are catching students' attention while building their academic skills. Information Gap is a curriculum-based routine that does just this. In this routine, students follow a structured pattern of asking questions and responding with answers in order to solve a problem. Throughout this ping-ponging conversation, students pose and answer questions, clarify what is asked and happening in a problem, build common understandings, and share experiences relevant to the topic to then be able to solve the math problem. For anyone who has done this routine, we know that it is challenging for students. By playing with a partner, the stakes are lowered and the process is valued alongside the answer.

The second strategy we can use is to "make it social". IM has been intentional about shifting from the traditional I do, we do, you do and instead has flipped the experience to be you do then we do, ending with the teacher and students stamping the learning. Changing this orientation allows students to come to conclusions for themselves and then build understanding with and from their peers. In addition, a few language and instructional routines also create a communal environment for learning. Stronger and Clearer Each Time has students think or write individually about a response, use a structured partner strategy with multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response. Collect and Display is another routine that stabilizes the fleeting language that students use during partner, small-group, or whole-class activities in order for a student's own output to be used as a reference in developing their mathematical language. The teacher listens for and scribes the student output using written words, diagrams, and pictures. This collected output can be organized, revoiced, or explicitly connected to other language in a display for all students to use.

Lastly we have opportunities to "storify." The brain is wired to remember stories and to use narrative structure to make sense of the world. All students learn content more effectively if they can create a coherent narrative about the topic or process presented. One of my favorite routines is Three Reads because it helps students reframe how they approach word problems. The first read is about students being able to make sense of the word problem in students' own language, without considering the numbers. Simply, what is happening in this situation? The second read is when students, who are familiar with the story from the first read, now attend to the numbers. In this way, students are able to understand the numbers in context to make sense of their relationship to each other in preparation for the third read, where students decide how they will approach finding the solution. Word problems are a way of storifying equations which is essential in preparation for courses like algebra, where students make sense of multiple representations (graphs, tables, equations, and writing expressions).

Hammond's three tools give us a framework for students to experience academic success, which is one of the three elements needed for a culturally responsive classroom. Maintaining cultural competence and developing a critical consciousness then becomes the work of the practitioner; this should be tailored for your students. Maintaining cultural competence means that teachers and classrooms need to be a place where students' whole selves are welcome. Consider the music played in the background while students are working, and the names and places referenced in word problems. Students need to be able to be themselves and be in an environment where excellence is seen as an extension of who they already are.

The last element of a culturally responsive classroom is building critical consciousness to challenge the status quo. Personally, I think this is one of the greatest gifts in teaching a culturally responsive math class. When students ask why something is important to learn, I can with confidence always answer that math is about all of the ways we can creatively solve problems. In math, we consider the connections between abstract understandings, explore multiple ways to approach an understanding, analyze information that is offered, and work collaboratively to find solutions. As educators, our work is to then offer opportunities for students to use these same skills to make sense of the world beyond the classroom. This means that our classrooms and schools need to be learning environments that can evolve and respond when challenged by students who are developing their critical thinking. As educators, we have to be open enough to realize that when we immerse our students in a culturally responsive learning environment, they will be looking to see those values reflected in the overall culture of the school.

Next Steps

Where do you see opportunities in IM K-12 curricula to gamify, storify, and make it social? What does it look like to maintain cultural competence within the math classroom? How can you support students in building critical consciousness?

Read more about culturally responsive teaching and IM curricula in the following blog posts:

You can also view the webinar IM K-5 Math as a Support for Culturally Responsive Pedagogy , with Dr. LaToya Byrd.

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Culturally Responsive Teaching and Math - IM CERTIFIED® BLOG

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Raise your hand if you have been perplexed by professional learning that told you your class needs to be culturally responsive, but left you completely unsure about what that means in a math class. Yup, been there, done that!

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

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By Ashli Black and Elisa Smith
Grade 8 is a year marked by shifts in mathematical focus. Where grades 6 and 7 introduce students to negative numbers and using them in operations, grade 8 introduces them to irrational numbers and using them with the Pythagorean Theorem. Where grades 6 and 7 dig into one-variable statistics, grade 8 begins the work with two-variable statistics. And where grades 6 and 7 emphasize ratios and proportional relationships, grade 8 turns its attention toward functions, particularly linear ones. For this last shift, the question becomes how do you go from a foundation in proportional relationships to an understanding of all types of linear relationships? Well, you start with geometry.

The Geometry of Lines

So let's focus on lines for a bit from a geometric perspective. A line drawn on the coordinate plane has features students can use to describe it, including its slope. From left to right what's going on with that line? Can that change be quantified? Students using IM 6-8 Math are positioned to examine these questions critically by starting the year with congruence and similarity. The study of similar triangles-slope triangles-builds foundational ideas about slope.

( https://curriculum.illustrativemathematics.org/MS/teachers/3/2/10/index.html )

For the diagonal line shown here, if one student picks two points and makes a slope triangle and another picks two points and makes a slope triangle, they will end up with similar triangles. How do they know they are similar? Well, they can pull from earlier thinking, developed in Unit 1 and the first half of Unit 2, and use a sequence of rigid transformations and a dilation. Or they can say that since the triangles have two pairs of congruent angles then they must be similar. Regardless of how students determine the similarity of the triangles, they can conclude that the vertical length divided by the horizontal length for any slope triangle of a line has the same value and this value is a way to quantify the slope of the line. (And by doing the vertical divided by the horizontal they get a number that gets greater for steeper looking lines, which is nice.)

Once a method for calculating the slope of a line is established, students gain a way to write an equation for the line based on this calculation.

( https://curriculum.illustrativemathematics.org/MS/teachers/3/2/11/preparation.html )

The slope of the line shown here is $\frac{1}{3}$. Since the slope between any two points on the line is the same, we can also say that for the points $(1,1)$ and $(x,y)$ the value of vertical length, $y-1$, divided by the horizontal length, $x-1$, must equal $\frac{1}{3}$. That is, $\frac{y-1}{x-1}=\frac{1}{3}$. From quotients of side lengths of similar triangles, we can make an equation and use it as a point-tester. If a point makes the equation true, it must be on the line.

All of this means that students start Unit 3: Linear Relationships with a robust geometric understanding of the slope of a line built from the work with slope triangles. But not all the pieces are there yet to be able to generalize to all linear relationships: slope has only been positive thus far, and linear relationships have only been proportional. Connections between rate of change, slope, and the constant of proportionality still need to be made.

From Ladybugs and Ants to Stacking Cups

To help students make the connections between rate of change, slope, and the constant of proportionality, let's return to contexts and start from the more familiar proportional relationships. The unit starts by having students attend to precision while they label axes, choose appropriate axes scales, and compare proportional relationships, such as the ladybug and ant who start at the same time but move at different paces, as shown here.

( https://curriculum.illustrativemathematics.org/MS/teachers/3/3/1/index.html )

, students connected the constant of proportionality to the steepness of a graph of a proportional relationship. Now, they change focus from the constant of proportionality to rate of change, which is about how the output changes at a constant rate with respect to the input. They also practice identifying the rate of change in graphs, tables, equations, and contexts for different proportional relationships.

To help students take the next step toward working with non-proportional linear relationships in the second section of the unit, they begin by stacking cups.

( https://curriculum.illustrativemathematics.org/MS/teachers/3/3/5/index.html )

Stacking cups has two variables for students to keep track of: number of cups and height of the stack. The context is meant to be straightforward so students can switch fluidly between the concrete situation and the abstract calculation for the rate of change and connect the two. Here the rate of change has a tangible meaning: add 1 cup to the stack, increase the height of the stack by a fixed amount. Add 2 cups, increase the height by double that fixed amount. Plotting the pairs of (number of cups in the stack, height in centimeters of the stack) results in a set of points that a single straight line can go through. The connection between the slope of the line, the rate of change, and the context is made. At the heart of Unit 3 is establishing that non-proportional linear relationships exist, that they can be graphed as a line with an associated slope, that they have a rate of change just like proportional relationships from grades 6 and 7, and that the value of the slope is the same as the value of the rate of change.

This last point is something remarkable. A connection between the quotients of sides of triangles and the change in output divided by the change in input. Whether students think geometrically to calculate the slope of a line or algebraically to calculate the rate of change, they will get the same value.

The next lessons offer a second way to approach linear relationships by expressing regularity in repeated calculations. Students start by connecting a set of graphs to their matching contexts while paying close attention to the meaning of the vertical intercept and the value of the slope ( Lesson 6, Activity 2 ). Only after this thinking do students consider how adding identical objects one after the other to a water-filled cylinder with a known initial height could be represented with an equation. It's here that the $y = mx + b$ form is introduced with $m$ as the rate of change and $b$ as the initial amount.

But let's not forget where all this started: geometry. The third way students conceptualize linear relationships is through transformations. Calling back to , any line in the plane can be considered a vertical translation of a line through the origin. This work happens through contexts, visually with graphs, and in thinking through the difference between equations $y = mx$ and $y = mx + b$.

Refining Thinking: Calculating Slope and Solutions to Linear Equations

Rounding out the unit, students take their first steps into working with contexts that have a negative rate of change, which in turn leads to refining how to calculate slope . They also consider what happens when one of the two variables does not vary while the other one can take any value, leading to vertical and horizontal lines. The final lessons of the unit continue the work of connecting contexts and their equations while purposefully thinking about what it means for a point to be a solution to an equation. This work builds a strong foundation for the next units on systems of equations, functions, and modeling scatter plots with lines of fit.

Next Steps

Unlocking the full range of linear relationships opens multiple avenues of study for the future, not least of which is that systems of equations in Unit 4 are far more interesting when you have more than just proportional relationships to consider. Where else can you see the work of this unit playing out? In particular, how does the work in Unit 3 making connections between multiple representations to reason about linear relationships set students up for success as they reason about linear functions in Unit 5?

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The Story of Grade 8 Unit 3: Linear Relationships - IM CERTIFIED® BLOG

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By Ashli Black and Elisa Smith Grade 8 is a year marked by shifts in mathematical focus. Where grades 6 and 7 introduce students to negative numbers and using them in operations, grade 8 introduces them to irrational numbers and using them with the Pythagorean Theorem.

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Imagine Learning has many supports embedded into the platform to support teachers in addressing student gaps.

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Unit 9 in IM Grades 6-8: Hidden Gems - IM CERTIFIED® BLOG

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By Lisa Matthews, IM Certified® Facilitator and PL Specialist Team for Grades 6-12 The IM curriculum is so thoughtfully designed and written that even those of us who have spent years with IM often find content that is new, or ideas within it that deepen our math teaching practice.

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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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Supporting Students with a Variety of Entry Points

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Supporting Students with a Variety of Entry Points

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In a class of 25 students, there are 25 different learners with 25 different backgrounds. Students come into the classroom with varied experiences with school, with mathematical content, and with life. While this has been true for time immemorial, teachers may feel it more acutely since the onset of the pandemic. The IM Team would like to support teachers with planning for the variety of student entry points. In particular, what features in the curriculum support strategies for providing additional time on prior content?

IM's Approach to the Diversity of Student Entry Points

IM recognizes that students approach new mathematical learning with different entry points. Some teachers may have experienced a wider variety of entry points due to disruptions to the 2019-2020 and 2020-2021 school year. Other teachers may consistently experience a wide range of entry points, for reasons including disruption in formal education, lack of full access to content in prior grades, or because some students take more time to fully know, use, and enjoy content.

Extensive remediation on below grade-level content is not productive. When students have difficulties accessing grade-level work, the approach of giving students work from earlier grades is ineffective, and approaches that provide access to grade-level work are preferred. A recent study by TNTP showed that giving students work from earlier grades results in losing more academic ground and getting even less access to grade-level work in the future. In other words, "meeting students where they are" and trying to remediate learning deficits often results in having to meet them even further back next year. Those stuck in this cycle are disproportionately black and brown students, students experiencing poverty, students with learning disabilities, and students learning English.

Learning new ways to support students coming in with different entry points is challenging, but ". . . trying to give every child a real chance to do grade-level work, however imperfectly, will lead to far better results than picking and choosing who gets those opportunities-right now and over the long run." ( Steiner and Weisberg, 2020 )

IM is designed to provide access and challenge to all students within the entire, inclusive classroom. All students benefit from being part of the conversations and hearing multiple perspectives and approaches from their peers, and the curriculum is designed with a wide variety of entry points in mind.

Teachers take advantage of built-in features of the curriculum to respond to student thinking during grade-level instruction, including:

accessible tasks designed to invite a variety of responses and give students a chance to learn from studying multiple approaches

warm-up activities and activity launches designed to provide an invitation to the mathematics and activate prior knowledge in the service of accessing grade-level content

prompts in each activity to address misconceptions

IM has several built-in features to position each teacher to respond to student thinking in real time. There are suggested adjustments based on student responses to cool-downs and pre-unit diagnostic assessment questions. There are also additional resources, such as the Algebra 1 Extra Support materials. When a teacher is responding to student thinking during the lesson, each student's learning is being addressed just in time and not postponed until later with a separate experience.

Strategies for Providing Additional Time on Prior Content
Given the wide variety of starting points students come in with, teachers may find that some students need additional time and additional instruction to be ready for grade-level content.
Schools provide additional time and additional instruction in a variety of ways, depending on student needs and available resources.
Support for the whole class
When most students in a class need some additional instruction to access grade-level content, teachers might use an optional lesson already in the curriculum, or use the suggestions in a Curriculum Adaptation Pack to find an activity, lesson, or practice problem from a previous grade or unit to supplement.

Adding supplemental material from a prior grade so that students can access grade-level content should be:

Just in time, rather than preventative. Rather than beginning the year with a long diagnostic assessment and spending several weeks covering prior-year material, use the Check Your Readiness pre-unit diagnostic questions and Curriculum Adaptation Packs to determine interventions needed just before encountering grade-level content within the unit.

Limited to what students need to access the grade-level content, so that students have a chance to do the major work of the grade. For example, in Grade 8 Unit 2, students will recognize when two slopes are equal. An unproductive approach would be to stretch back to grade 4 work on fractions of a pizza, or fractions of a dollar, or other contexts focused on visualizing equivalent fractions-this is unhelpful for the purposes of comparing slopes. Instead, the work students have just been doing in Unit 2 on similar triangles will help them in Unit 3 recognize that, for example, the slope 4/3 and the slope 8/6 are equal, and they can use different arguments for explaining why. A quick refresher on equal quotients, such as extending the warm up from Grade 8 Unit 2 Lesson 10 to include some more examples and a discussion of different strategies, some references posted in the room, and having a calculator handy may be all that is needed.

Balanced with thoughtful prioritization. IM used the guidance from Student Achievement Partners to prioritize major work of the grade in recommending lessons and topics to cut if most students need significant additional time to be ready for the grade. This ensures that students get to experience instruction on major work of the grade and don't fall further behind.

Support for individual students

Unfinished learning is not the same as special education or having an Individualized Education Plan. Sometimes these two are conflated, but many students who have unfinished learning do not have learning disabilities, and many students with learning disabilities are ready for grade-level content provided there are multiple means of access and engagement. IM's support for students with disabilities is designed to provide those multiple means per the Universal Design for Learning principles so each student has access to the grade-level content in the curriculum.

Because of this, students may need additional time focused on math for multiple reasons:

A student with an IEP may have the prerequisite understanding and skills, but may need more time to process the grade-level content being worked on or more ways to explore the content.

A student may be able, generally, to learn in the amount of time and structure of their math class but may need to understand additional content or skills from prior grades in order to be successful with upcoming content.

A student may need to learn content that they need to access upcoming content and may need more time and more means of access and engagement, representation, and action and expression to learn that content and the grade-level content being taught in their math class.

Therefore, it is important to be clear in planning whether additional time outside of math class is designed to make sure students are ready with prerequisite knowledge, or to give students who are ready for grade-level content but need more time and a wider variety of means of accessing the content, or an intervention class is serving both needs at once.

The IM curriculum does not currently have an aligned intervention product. However, when there is time outside the math block for additional support for individual students, we recommend fortifying the representations and routines from IM rather than learning other representations and routines. Being introduced to too many strategies, programs, and representations causes cognitive overload and dilutes students' ability to focus on the math content. Extra time with what a student is seeing in math class, and with just-in-time prerequisite content, gives opportunities to be more successful in math class. For the Algebra 1 course, there is a set of support materials designed for this purpose. Read more about the Algebra 1 Extra Support materials.

In order to facilitate providing additional support outside of the math block with aligned routines, representations, approaches, etc., IM has resources that schools and districts can take advantage of:

pre-unit diagnostic problems (included in IM 6-12 Math as the Check Your Readiness assessment). These can aid in determining which students may need additional time focused on prerequisite content as well as identifying the most important prerequisite content

optional lessons and activities that often provide extra practice or review prior grade-level materials

notes for teachers in each cool-down with suggestions for supporting students including suggestions for reviewing prior learning to give more students access to the grade-level content. These can help teachers identify opportunities for both supporting grade-level content and supplementing with work on prerequisite content that may be impacting students' current work

Curriculum Adaptation Packs, available on the LearnZillion platform and in the IM Community Hub, which suggest prior lessons, activities, and sections to revisit to help students get ready for upcoming work in a "just in time" fashion

Next Steps

Reflect: how do you support various student access points during math class? What structures can you put in place outside of class to support student access during class, and what might these additional math blocks look like? What parts of the IM curriculum would you want to use or reinforce?

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What Does IM Think About Unfinished Learning and Supporting Students with a Variety of Entry Points? (Grades 6-12) - IM CERTIFIED® BLOG

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In a class of 25 students, there are 25 different learners with 25 different backgrounds. Students come into the classroom with varied experiences with school, with mathematical content, and with life. While this has been true for time immemorial, teachers may feel it more acutely since the onset of the pandemic.

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Many Spanish resources are available for students and parents.

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By Lisa Matthews, IM Certified® Facilitator and PL Specialist Team for Grades 6-12 The IM curriculum is so thoughtfully designed and written that even those of us who have spent years with IM often find content that is new,...

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Here is an example of a digital applet that is used to explore the addition and subtraction of integers. Digital applets are often used to support student exploration as they learn new topics.