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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.
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.
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.
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.
The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics
(https://im612.org/k12numberline)
In this article from the American Mathematical Society (https://im612.org/AMS), 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: 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).
The Secret Life of the \(ax + b\) Group
(https://im612.org/secretlifeaffinegroup)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Howe delves into the ways in which the affine group of the line (the \(ax + b\) group) surfaces in high school mathematics, and discusses some extensions of standard topics suggested by this point of view.
Different Uses of Letters and Types of Representations in Algebra
(https://im612.org/lettersmathematics)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Oehrtman and Shultz explore the uses of letters (variables, constants, parameters) in high school mathematics and beyond.
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.
Inverses
(https://im612.org/inverses)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Farrand-Shultz unifies the different contexts in which “inverses” are used in mathematics, including the multiplicative inverse of a number, the inverse of a function, and the inverse relationship between differentiation and antidifferentiation. Many of these notions can be unified by noticing the operation relative to which the inverse is defined.
To learn more about the progression of statistical concepts through high school and beyond, see:
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).
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.
Extraneous and Lost Roots
(https://im612.org/lostroots)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), McCallum characterizes solving an equation as applying a function to each side of the equation, and the conditions that result in extraneous roots.
To learn more about the progression of statistical concepts through high school and beyond, see:
To learn more about the progression of function concepts through high school and beyond, see: Progressions for the Common Core State Standards in Mathematics, High School Functions (https://imk12.org/8hsfprogression).
The image of function in school mathematics
(https://im612.org/functionschoolmath)
In this blog post, McCallum explores the variation in representations of functions in school mathematics in the U.S. versus other countries, and highlights the ways in which some representations can be problematic.
Equations and Functions, Variables and Expressions
(https://im612.org/equationsfunctionsconnected)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Howe elaborates on the notions of equations and functions, shows how they are connected, and illustrates how to use them to express many ideas.
The Equal Sign, Equations, and Functions
(https://im612.org/equationsfunctionsdifferent)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Friedberg seeks clarity on how equations and functions are different concepts, even though equations are used to describe functions.
3 Views of F(x, y) = 0
(https://im612.org/equationsfunctionsuses)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Stanley explores differing uses of the terms equation and function in the fields of analysis, algebra, and analytic geometry.
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 Algebra 2 depicts the exponential growth in population of two colonies of bacteria, measured by the area of the colony.
How Math (and Vaccines) Keep You Safe From the Flu
(https://im612.org/mathpreventepidemic)
In this article, Honner shows how widespread vaccination can disrupt the exponential spread of disease and prevent epidemics. Simple epidemiological concepts, such as R0, are introduced.
How do I Solve this Equation? Look at the Symmetries! — The Idea behind Galois Theory
(https://im612.org/galoistheory)
In this vignette from the Klein Project (https://im612.org/kleinprojectblog), author Timo Leuders explains, “The attempts in history to find a general solution procedure for polynomial equations finally lead to a transformation of classical algebra (as the art of solving equations) into modern algebra (as the analysis of structure and symmetry). A culmination point in this development was the work of Évariste Galois (1811–1832). This paper tries to give a less technical account of Galois’ ideas that changed algebra by showing examples that highlight what it means to ‘look at structure and symmetry’ when trying to solve equations.”
To learn more about the progression of geometrical concepts through high school and beyond, see: Progressions for the Common Core State Standards in Mathematics, Geometry 7–8 and High School (https://imk12.org/7hsgprogression).
Proof in IM’s High School Geometry (A Sneak Preview)
(https://im612.org/proofimhsgeometry)
Rigor in Proofs
(https://im612.org/rigorinproofs)
In these two blog posts, the authors (Ray-Riek and Cardone) and (Cardone and Rosenberg) situate the treatment of learning how to write proofs in IM Geometry within a structure that appears in different forms throughout a math learner’s education, including before and beyond high school mathematics.
Triangle Congruence and Similarity: A Common-Core-Compatible Approach
(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.
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, which is the topic of a later unit in this course.
Making Peace with the Basics of Trigonometry
(https://im612.org/makingpeacetrig)
In this post, Phillips explains why the ratios are presented in the order cosine, sine, and tangent to set students up for greater success in future coursework.
The Math of Social Distancing Is a Lesson in Geometry
(https://im612.org/mathsocialdistancing)
In this article, Honner takes a look at some of the surprising complications that arise when trying to pack a plane with circles or a three-dimensional space with spheres. Circle- and sphere-packing play a part in modeling crystal structures in chemistry and abstract message spaces in information theory. It’s a simple-sounding problem that’s occupied some of history’s greatest mathematicians, and exciting research is still happening today, particularly in higher dimensions.
What is the way of packing oranges? — Kepler’s conjecture on the packing of spheres
(https://im612.org/packingoranges)
In this vignette from the Klein Project (https://im612.org/kleinprojectblog), Rousseau provides a proof of the densest packing of discs in two-dimensional space and spheres in three-dimensional space.
What is a straight line?
(https://im612.org/straightline)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Chazan and McCallum relate Euclid’s definition of a straight line to two other ways of describing straight lines.
How Google works: Markov chains and eigenvalues
(https://im612.org/googlemarkovchains)
In this vignette from the Klein Project (https://im612.org/kleinprojectblog), Rousseau applies concepts of probability to illuminate how Google’s PageRank algorithm works.
The Secret Life of the \(ax + b\) Group
(https://im612.org/secretlifeaffinegroup)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Howe delves into the ways in which the affine group of the line (the \(ax+b\) group) surfaces in high school mathematics, and discusses some extensions of standard topics suggested by this point of view.
Different Uses of Letters and Types of Representations in Algebra
(https://im612.org/lettersmathematics)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Oehrtman and Shultz explore the uses of letters (variables, constants, parameters) in high school mathematics and beyond.
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.
Inverses
(https://im612.org/inverses)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), Farrand-Shultz unifies the different contexts in which “inverses” are used in mathematics, including the multiplicative inverse of a number, the inverse of a function, and the inverse relationship between differentiation and antidifferentiation. Many of these notions can be unified by noticing the operation relative to which the inverse is defined.
How do I Solve this Equation? Look at the Symmetries! — The Idea behind Galois Theory
(https://im612.org/galoistheory)
In this vignette from the Klein Project (https://im612.org/kleinprojectblog), author Timo Leuders explains, “The attempts in history to find a general solution procedure for polynomial equations finally lead to a transformation of classical algebra (as the art of solving equations) into modern algebra (as the analysis of structure and symmetry). A culmination point in this development was the work of Évariste Galois (1811–1832). This paper tries to give a less technical account of Galois’ ideas that changed algebra by showing examples that highlight what it means to ‘look at structure and symmetry’ when trying to solve equations.”
Algebraic Number Starscapes
(https://im612.org/algebraicstarscapes)
In this paper, the authors Harriss, Stange, and Trettel describe how they used computer visualization to study the geometry of numbers in the complex plane and their Diophantine approximation. The authors explain: “Motivated by the resulting images, which we have called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases.”
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.
Extraneous and Lost Roots
(https://im612.org/lostroots)
In this essay from the Noyce-Dana project (https://im612.org/noycedanaproject), McCallum characterizes solving an equation as applying a function to each side of the equation, and the conditions that result in extraneous roots.
Calculators, Power Series, and Chebyshev Polynomials
(https://im612.org/powerseries)
In this vignette from the Klein Project (https://im612.org/kleinprojectblog), Cohen introduces the concept of a power series, which can be thought of as a polynomial function of infinite degree, and, second, shows their application to evaluating functions on a calculator. The article, in particular, deduces a power series for sin x, and shows how to improve on the straightforward approach to approximating its values.
To learn more about the progression of statistical concepts through high school and beyond, see:
Progressions for the Common Core State Standards in Mathematics, High School Statistics and Probability (https://imk12.org/hsspprogression).
Peck, R., Gould, R. and Miller, S. (2013). Developing essential understanding in statistics for grades 9–12. National Council of Teachers of Mathematics.