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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.
Universal Design for Learning
Visualizing IM K–5 Math™ within a Dream Team of Supports (https://imk5.org/visualizingk5part1), Visualizing IM K–5 Math in Specialized Academic Settings: Part 2 (https://imk5.org/visualizingk5part2), and Visualizing IM K–5 Math in Specialized Academic Settings: Part 3 (https://imk5.org/visualizingk5part3)
In this three-part blog series, Jimenez and Mouton focus on meeting the needs of all students engaged in Tier 1 core curriculum. Part 1 outlines IM’s intentional integration of UDL principles in the K–5 curriculum. Part 2 explores how to integrate IM lessons in a specialized academic instructional setting. Part 3 unpacks the integration of instructional routines, centers, and representations from IM K–5 Math into intensive academic support settings.
Enacting IM K–5 Math™ Lessons in a Grade 4 and 5 Special Education Class (https://imk5.org/enactinglessonsg4g5). In this blog post, Guarino and Huerta respond to the question, “What does it look like for everyone in a school to learn together?” Learn more about the embedded IM K–5 Math strategies used to uplift joy and classroom energy for all students.
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.
K-5 Curriculum Design Features that Support Equity and Inclusion (https://imk5.org/designsupportequityinclusion). In this blog post, Aminata highlights features of the IM K–5 curriculum that mirror the culturally responsive lesson structure in Zaretta Hammond’s book, Culturally Responsive Teaching and the Brain.
The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics (https://imk12.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 K–12, see Progressions for Common Core State Standards in Mathematics, K–12 Modeling (https://imk12.org/k12mprogression).
Units, a Unifying Idea in Measurement, Fractions, and Base Ten (https://imk5.org/unifyingidea). In this blog post, Zimba illustrates how units “make the uncountable countable,” and discusses how the foundation—built around structuring space, in K–2 measurement and geometry—allows for the development of fractional units and beyond to irrational units.
Untangling fractions, ratios, and quotients. (https://imk5.org/untangle) In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.
The Progressions for the Common Core State Standards in Mathematics 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 documents that align with each unit in the K–5 materials for further reading.
| K | 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|---|
| Counting and Cardinality, K (https://imk12.org/kccprogression) | Units 1, 2, 4, and 6 | – | – | – | – | – |
| Operations and Algebraic Thinking, K–5 (https://imk12.org/k5oaprogression) | Units 4 and 5 | Units 1 and 2 | Unit 1 | Units 1 and 4 | Units 1 and 5 | Units 1 and 7 |
| Number and Operations in Base Ten, K–5 (https://imk12.org/k5nbtprogression) | Unit 6 | Units 3-5 | Units 2, 5, and 7 | Unit 3 | Units 4 and 6 | Units 4–6 |
| Number and Operations, Fractions, 3–5 (https://imk12.org/35nfprogression) | – | – | – | Unit 5 | Units 2 and 3 | Units 2, 3, and 5 |
| Measurement and Data, K–5 (https://imk12.org/k5mdprogression) | Unit 3 | Unit 1 | Unit 1 | Units 1 and 6 | Unit 3 | Unit 6 |
| Geometric Measurement, K–5 (https://imk12.org/k5gmprogression) | Unit 7 | Unit 6 | Unit 3 | Units 2, 6, and 7 | Unit 7 | Unit 1 |
| Geometry, K–6 (https://imk12.org/k6gprogression) | Units 3 and 7 | Unit 7 | Unit 6 | Unit 7 | Unit 8 | Unit 7 |
Unit 1
When is a number line not a number line?(https://imk5.org/notanumberline) In this blog post, McCallum shares why the number line is introduced in Grade 2 in IM K–5 Math, emphasizing the importance of foundational counting skills in IM Kindergarten.
Unit 7
What is a Measurable Attribute?(https://imk5.org/measurableattribute) In this blog post, Umland wonders what counts as a measurable attribute, and discusses how this interesting and important mathematical idea begins to develop in kindergarten.
Unit 2
The Power of Small Ideas.(https://imk5.org/smallideas) In this blog post, McCallum discusses, among other ideas, the use of a letter to represent a number. The foundation of this idea is introduced when students first represent an unknown with an empty box.
Representing Subtraction of Signed Numbers: Can You Spot the Difference? (https://imk5.org/Representingsubtraction) In this blog post, Anderson and Drawdy discuss how counting on to find the difference plays a foundational role in understanding subtraction with negative numbers, on the number line, in middle school.
Unit 3
Russell, S. J., Schifter D., & Bastable, V. (2011). Connecting arithmetic to algebra: Strategies for building algebraic thinking in the elementary grades. Heinemann. This book explains how generalizing the basic operations, rather than focusing on isolated computations, strengthens fluency and understanding, which helps prepare students for the transition from arithmetic to algebra. Chapter 1, Generalizing in Arithmetic, (https://imk5.org/generalizing) is available as a free sample from the publisher.
Unit 4
Rethinking Instruction for Lasting Understanding: An Example. (https://imk5.org/Rethinkinginstruction) In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.
Unit 4
To learn more about the essential nature of the number line (introduced in this unit) in mathematics beyond grade 2, see:
Unit 8
What is Multiplication? (https://imk5.org/Whatismultiplication) In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of IM Grade 2.
Unit 1
Ratio Tables are not Elementary. (https://imk5.org/notelementary) In this blog post, McCallum discusses the difference between multiplication tables and tables of equivalent ratios, highlighting how K–5 arithmetic work prepares students to make sense of these tables.
Unit 3
To learn more about the order of operations, see:
Unit 5
Fractions: Units and Equivalence. (https://imk5.org/fractions) In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.
Unit 2
Fractions: Units and Equivalence. (https://imk5.org/fractions) In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.
Unit 7
Making Peace with the Basics of Trigonometry. (https://imk12.org/peacewithtrig) In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.
Unit 1
To learn more about the order of operations, see:
Unit 3
Why is a negative times a negative a positive? (https://imk5.org/negative) In this blog post, McCallum discusses how the “rule” for multiplying negative numbers is grounded in the distributive property.
Units 4–6
To learn more about the progression of the number system through middle school and beyond, see the Progressions for the Common Core State Standards in Mathematics, The Number System, 6–8 and High School, Number (https://imk12.org/6hsnprogression).
Unit 7
To learn more about the progression of ratios and proportional reasoning through middle school and beyond, see the Progressions for the Common Core State Standards in Mathematics, 6–7, Ratios and Proportional Relationships (https://imk12.org/67rpprogression).
Making Sense of Distance in the Coordinate Plane. (https://imk5.org/makingsense) In this blog post, Richard shares how understanding of the coordinate plane, introduced in grade 5, provides a foundation for conceptual understanding of distance and the Pythagorean Theorem.
© Illustrative Mathematics 2021. Released under a CC BY 4.0 International License. Cited works remain under their original respective licenses.