These articles, curated by the Illustrative Mathematics curriculum team, explain how concepts extend beyond the target grade level and provide guidance to enhance grade-level access and inclusivity for diverse learners. Please note that these articles vary in their scope, with focus ranging between specific units, a full course, and the entire series.
Enhancing Access
Universal Design for Learning
- Empowering All Storytellers: Tips for Engaging Special Populations Using IM® v.360 for Grade 6–12
(https://im612.org/engagingspecialpopulations)
In this blog post, authors Cardone and Jimenez unpack creative solutions to meet the needs of secondary special education students in single teacher and co-teacher math learning environments.
Multilingual Learners
- Tackling Wordy Problems: How the Three Reads Math Language Routine Supports Access for All Learners
(https://im612.org/tacklingwordyproblems)
In this blog post, Herbert offers guidance on implementing the Three Reads math language routine to amplify, rather than simplify, language and provide access to mathematics to all students.
- Think Pair Share
(https://imk12.org/thinkpairshareblog)
In this blog post, Wilson explains how Think-Pair-Share makes space for extra processing time, which is particularly beneficial for English learners and learners with disabilities. This post includes classroom footage of the routine and students’ reflections about their experience.
- Making Sense of Story Problems
(https://imk5.org/makingsensestoryproblems)
In this blog post, Peart outlines how the IM K–5 Math curriculum writing team built in student support to make sense of word problems through relevant contexts, routines such as Act It Out and Three Reads, and visual representations.
- Math Language Routines: Discourse with a Purpose
(https://imk12.org/mlrdiscoursewithpurpose)
In this blog post, Taylor provides an overview of the eight math language routines, their connections to the Standards for Mathematical Practice, and how their use in the IM Curricula helps students advance their thinking.
- Unlocking Learners’ Thinking Using the Mathematical Language Routines
(https://imk12.org/unlockingthinkingmlrs)
In this blog post, Fricchione and Rundstrom unpack how the math language routines Collect and Display, Compare and Connect, and Discussion Supports help English learners participate fully in the math classroom while acquiring English.
- A Circumference By Any Other Name . . .
(https://imk12.org/byanyothername)
In this blog post, Phillips lays out the case for waiting to introduce vocabulary until after students have a chance to explore a concept.
Culturally Responsive and Sustaining Education
- Cultivating Joy in the IM Classroom
(https://imk12.org/cultivatingjoy)
In this blog post, Peart offers practical tips for using the curriculum and its components to create a classroom environment where all students know, use, and enjoy mathematics.
- Culturally Responsive Teaching and Math
(https://imk12.org/culturallyresponsiveteaching)
In this blog post, Howlette outlines how to use the features in IM K–12 Math to support and sustain culturally responsive education in the classroom.
Entire Series
- The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics.
(https://im612.org/k12numberline)
In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers, from kindergarten to grade 12, and address the work that students might do in later years.
- To learn more about the progression of modeling concepts through high school and beyond, see the Progressions for the Common Core State Standards in Mathematics, K–12 Modeling (https://imk12.org/k12mprogression). To learn more about the use of modeling prompts in the Illustrative Mathematics high school materials, see Making Authentic Modeling Possible (https://im612.org/authenticmodelingpossible)
- To learn more about the progression of the study of units and quantity through high school and beyond, see: Progressions for the Common Core State Standards in Mathematics, High School Quantity (https://imk12.org/hsqprogression).
- Truth and Consequences Revisited.
(https://im612.org/truthandconsequences)
In this blog post, McCallum describes what happens when trying to solve equations (that are not linear, in future units and courses), with steps that don’t have true converses.
- What Is a Variable?
(https://im612.org/whatisavariable)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Epp traces the different uses of variables in high school mathematics and beyond by drawing parallels to the use of pronouns in written and spoken English.
- To learn more about the progression of algebra concepts through high school and beyond, see: Progressions for the Common Core State Standards in Mathematics, High School Algebra (https://imk12.org/hsaprogression).
CCSS Progressions
The Progressions for the Common Core State Standards describe the progression of a topic across grade levels, note key connections among standards, and discuss challenging mathematical concepts. This table provides a mapping of the particular progressions that align with each unit in the materials for IM Grades 6–8 Accelerated, for further reading.
|
Chapters
|
Accelerated 6
|
Accelerated 7
|
|
Geometry, K–6 (https://imk12.org/k6gprogression)
|
Units 1 and 3 |
– |
|
Ratios and Proportional Relationships, 6–7 (https://imk12.org/67rpprogression)
|
Units 2, 5, and 6 |
– |
|
Expressions and Equations, 6–8 (https://imk12.org/68eeprogression)
|
Units 4 and 7 |
Units 2–5, 7, 8 |
|
Statistics and probability, 6–8 (https://imk12.org/68spprogression)
|
Unit 8 |
Unit 5 |
|
The Number System, 6–8 (https://imk12.org/6hsnprogression)
|
Units 3 and 7 |
Unit 8 |
|
Geometry, 7–8 (https://imk12.org/7hsgprogression)
|
Unit 5 |
Units 1, 2, 6, 8 |
|
Functions, 8 (https://imk12.org/8hsfprogression)
|
– |
Units 5 and 6 |
Accelerated 6
Entire course
Unit 2
- Why We Don’t Cross Multiply.
(https://im612.org/whywedontcrossmultiply)
In this blog post, Nowak and Gray discuss how the IM Curriculum develops students’ understanding of equivalent ratios and proportional relationships.
Unit 3
Fraction Division Parts 1–4. In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions, with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems.
Unit 7
- Why is 3 – 5 = 3 + (–5)?
(https://im612.org/numberlinenegativenumbers)
In this blog post, McCallum discusses the use of the number line in introducing negative numbers.
- Why is a negative times a negative a positive?
(https://im612.org/negativenegativepositive)
In this blog post, McCallum discusses how the “rule” for multiplying negative numbers is grounded in the distributive property.
- Rethinking Instruction for Lasting Understanding: An Example.
(https://im612.org/lastingunderstanding)
In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.
- To learn more about the progression of the study of number systems through high school and beyond, see: Progressions for the Common Core State Standards in Mathematics, High School Number (https://imk12.org/6hsnprogression)
Unit 8
- A Thread Through Early Algebra 1.
(https://im612.org/earlyalgebra1)
In this blog post, Petersen and Black illustrate how the statistics work in middle school sets students up for success in Algebra 1.
To learn more about the progression of statistical concepts in high school and beyond, see:
- Progressions for the Common Core State Standards in Mathematics, High School Statistics and Probability.
(https://imk12.org/hsspprogression)
- Kader, G., Jacobbe, T., Wilson, P., Zbiek, R. M. (2013). Developing essential understanding of statistics for teaching mathematics in grades 6–8. National Council of Teachers of Mathematics.
Accelerated 7
Entire Course
- Coherence between Grade 8 and Algebra 1.
(https://im612.org/coherenceg8alg1)
In this blog post, Ortega outlines some key differences between the topics of seeing structure, solving systems of equations, and functions, when looking at IM Grade 8 and IM Algebra 1 content.
Unit 1
- Triangle Congruence and Similarity: A Common-Core-Compatible Approach Criteria.
(https://im612.org/tricongruencesimilarity)
This article by Douglas and Picciotto (https://im612.org/mathed) walks through an approach to triangle congruence and similarity that is consistent with basing the study of these ideas on geometric transformations, which extends to the similarity of curves.
- Geometry Through the Eyes of Felix Klein.
(https://im612.org/geometryklein)
In this excerpt from Visual Complex Analysis (https://im612.org/visualcomplexanalysis), Needham notes that geometric congruence in the plane is an equivalence relation, and that a family of transformation, which satisfies certain requirements, is a group.
- Proof in IM’s High School Geometry (A Sneak Preview).
(https://im612.org/proofimhsgeometry)
In this blog post, Ray-Riek and Cardone describe the expectations for proof writing in high school geometry.
Unit 3
- Rethinking Instruction for Lasting Understanding: An Example.
(https://im612.org/lastingunderstanding)
In this blog post, Nowak offers insights about helping students develop understanding of solving inequalities.
Unit 5
To learn more about the progression of statistical concepts in high school and beyond, see:
- Progressions for the Common Core State Standards in Mathematics, High School Statistics and Probability.
(https://imk12.org/hsspprogression)
- Kader, G., Jacobbe, T., Wilson, P., Zbiek, R. M. (2013). Developing essential understanding of statistics for teaching mathematics in grades 6–8. National Council of Teachers of Mathematics.
Unit 6
- Reading Graphs is a Complex Skill.
(https://im612.org/readinggraphs)
In this blog post, McCallum traces the skill of reading graphs, with examples from IM Grade 5, IM Grade 8, and IM Algebra 2. The example from IM Grade 8 depicts a graph students use to describe the functional relationship between two quantities.
- The image of function in school mathematics.
(https://im612.org/functionschoolmath)
In this blog post, McCallum explores the variation in representations of functions in U.S. school mathematics in the U.S. versus other countries, and highlights the ways in which some representations can be problematic.
Unit 8
- Making Sense of Distance in the Coordinate Plane.
(https://im612.org/distancecoordinateplane)
In this post, Richard shows how the distance formula is connected to an equation of a circle. Computing distance on the coordinate plane is introduced in this unit, and students build on these ideas, along with those of circles, in future courses.