## About These Materials

These materials were created by Illustrative Mathematics. They were piloted and revised in the 2019-2020 and 2020-2021 school years.

Each grade level contains 8 or 9 units. Units contain between 8 and 28 lesson plans. Each unit, depending on the grade level, has pre-unit practice problems in the first section, checkpoints or checklists after each section, and an end-of-unit assessment. In addition to lessons and assessments, units have aligned center activities to support the unit content and ongoing procedural fluency.

The time estimates in these materials refer to instructional time. Each lesson plan is designed to fit within a class period that is at least 60 minutes long. Some units contain optional lessons and some lessons contain optional activities that provide additional student practice for teachers to use at their discretion.

Teachers can access the teacher materials either in print or in a browser as a digital PDF. When possible, lesson materials should be projected so all students can see them.

Many activities require blackline masters of recording sheets, game boards, or cards that teachers need to photocopy or cut up ahead of time. Teachers might stock up on two sizes of resealable plastic bags: sandwich size and gallon size. For a given activity, one set of cards can go in each small bag, and then the small bags for one class can be placed in a large bag. If these are labeled and stored in an organized manner, it can facilitate preparing ahead of time and reusing card sets for multiple activities.

There are a few blackline masters that we recommend teachers copy once and laminate for frequent use. These blackline masters are provided as a Material to Copy in each grade in the first lesson or center in which they appear with a note, such as: “Number Cards are provided as a blackline master. Students will continue to use these throughout the year. Consider copying them on cardstock or laminating them and keeping them organized to be used repeatedly.” When these materials are needed again in that grade in lessons or centers, they are listed as a Material to Gather and the blackline master is not provided.

This list shows where these blackline masters first appear in each grade, when applicable.

Counting Mats:

- Kindergarten, Unit 1, Lesson 12

5-frames:

- Kindergarten, Unit 1, Lesson 3

10-frames:

- Kindergarten, Unit 5, Lesson 13
- Grade 1, Unit 1, Lesson 1
- Grade 2, Unit 1, Lesson 17

Double 10-frames:

- Grade 1, Unit 3, Lesson 9

Number Cards (0–10):

- Kindergarten, Unit 2, Lesson 14
- Grade 1, Unit 1, Lesson 2
- Grade 2, Unit 1, Lesson 1
- Grade 3, Unit 3, Suggested Center, How Close? Stage 4
- Grade 4, Unit 1, Suggested Center, Can You Build It Stage 2
- Grade 5, Unit 1, Suggested Center, Can You Build It Stage 3

### Further Reading

The curriculum team at Illustrative Mathematics has curated some articles that contain adult-level explanations and examples of where concepts lead beyond the indicated grade level. These are recommendations that can be used as resources for study to renew and fortify the knowledge of elementary mathematics teachers and other educators.

### Entire Series

The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics. 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 the Progressions for Common Core State Standards in Mathematics, K–12 Modeling.

### K–2

Units, a Unifying Idea in Measurement, Fractions, and Base Ten. In this blog post, Zimba illustrates how units “make the uncountable countable” and discusses how the foundation built in K–2 measurement and geometry around structuring space allows for the development of fractional units and beyond to irrational units.

### 3–5

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.- Fraction Division Part 1: How do you know when it is division?
- Fraction Division Part 2: Two interpretations of division
- Fraction Division Part 3: Why invert and multiply?
- Fraction Division Part 4: Our final post on this subject (for now)

Untangling fractions, ratios, and quotients. In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.

### CCSS Progressions Documents

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 documents that align with each unit in the K–5 materials for further reading.

K | 1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|---|

Counting and Cardinality and Operations and Algebraic Thinking | Units 1, 2, 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 | Unit 6 | Units 3-5 | Units 2, 5, and 7 | Unit 3 | Units 4 and 6 | Units 4–6 |

Number and Operations, Fractions | – | – | – | Unit 5 | Units 2 and 3 | Units 2, 3, and 5 |

Data | Unit 3 | Unit 1 | Unit 1 | Units 1 and 6 | Unit 3 | Unit 6 |

Geometric Measurement | Unit 7 | Unit 6 | Unit 3 | Units 2, 6, and 7 | Unit 7 | Unit 1 |

Geometry | Units 3 and 7 | Unit 7 | Unit 6 | Unit 7 | Unit 8 | Unit 7 |

### Kindergarten

**Unit 1**

When is a number line not a number line? 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.

**Unit 7**

What is a Measurable Attribute? In this blog post, Umland wonders about what counts as a measurable attribute and discusses how this interesting and important mathematical idea begins to develop in kindergarten.

### Grade 1

**Unit 2**

The Power of Small Ideas. 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 in this unit when students first represent an unknown with an empty box.

Representing Subtraction of Signed Numbers: Can You Spot the Difference? 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 students’ fluency and understanding which helps prepare them for the transition from arithmetic to algebra. Chapter 1, Generalizing in Arithmetic, is available as a free sample from the publisher.

**Unit 4**

Rethinking Instruction for Lasting Understanding: An Example. In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.

### Grade 2

**Unit 4**

To learn more about the essential nature of the number line (which is introduced in this unit) in mathematics beyond grade 2, see:

- The Nuances of Understanding a Fraction as a Number. In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers.
- Why is 3 – 5 = 3 + (-5)? In this blog post, McCallum discusses the use of the number line in introducing negative numbers.

**Unit 8**

What is Multiplication? In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.

### Grade 3

**Unit 1**

Ratio Tables are not Elementary. 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:

- A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.
- Notes on Order of Operations. In this essay from the Noyce-Dana project, Hsu and Madden discuss the order of operations for both arithmetic and algebraic expressions.

**Unit 5**

Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.

### Grade 4

**Unit 2**

Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.

**Unit 7**

Making Peace with the Basics of Trigonometry. 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.

### Grade 5

**Unit 1**

To learn more about the order of operations, see:

- A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.
- Notes on Order of Operations. In this essay from the Noyce-Dana project, Hsu and Madden discuss the order of operations for both arithmetic and algebraic expressions.

**Unit 3**

Why is a negative times a negative a positive? 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.

**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.

Making Sense of Distance in the Coordinate Plane. 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.