The number of days includes two assessment days per unit. The upper bound of the range includes optional lessons.

 

In the unit dependency chart, an arrow indicates that a particular unit is designed for students who already know the material in a previous unit. Reversing the order of the units would have a negative effect on mathematical or pedagogical coherence.

The following chart shows unit dependencies across the curriculum for IM Grades 3–8.

Section Dependency Diagrams

In the section dependency charts, an arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.

IM v.360 organizes each grade level into eight or nine units that each address a major concept and a group of related standards. The unit titles communicate the major concepts that are covered in each grade level.

One way to visualize the content connections is by mapping where each domain of the California Common Core State Standards for Mathematics is covered in IM v.360. Each unit generally addresses several related standards within a primary domain, while also making connections to relevant standards in other supporting domains. This structure supports the vertical alignment of the curriculum as a whole. The domain connections diagrams for each grade band demonstrate that the architecture of the curriculum considers more than simply covering individual standards one by one.

All domain-related information has been coded by color/shape to demonstrate similarities and progressions across grade bands. The key is as follows:

• Blue Rhombus:
  • K only: Counting & Cardinality (CC)
• Red Triangle:
  • K–5: Operations & Algebraic Thinking (OA)
  • 6–8: Expressions and Equations (EE)
  • 9–12: HS Algebra (HSA)
• Yellow Rectangle:
  • K–5: Numbers and Operations in Base Ten (NBT)
  • 6–8: The Number System (NS)
  • 9–12: HS Number and Quantity (HSN)
• Orange Pentagon:
  • K–5: Numbers & Operations—Fractions (NF)
  • 6–7: Ratios & Proportional Relationships (RP)
  • 8: Functions (F)
  • 9–12: HS Functions (HSF)
• Purple Hexagon:
  • K–5: Measurement & Data (MD)
  • 6–8: Statistics & Probability (SP)
  • 9–12: HS Statistics & Probability (HSS)
• Green Circle:
  • All grade levels: Geometry (G/HSG)

Primary Domains

This chart shows the primary domain addressed in each unit. Other supporting domains are not shown here but are shown in the charts below.

Counting and Cardinality

The Counting and Cardinality (CC) domain, while specific to Kindergarten, underlies the Operations and Algebraic Thinking (OA) domain as well as Numbers and Operations in Base Ten (NBT). It is foundational for students’ work in subsequent grades.

K: The CC domain is present in every unit within the Kindergarten course and prepares students for the work they will do throughout Grade 1. These skills are essential for students to grasp concepts that emphasize the NBT and OA domains, such as adding and subtracting within 20 and the understanding of numbers to 99. Here, we’ve highlighted the domain of each unit in IM v.360 Grade 1 that is most affected by the Counting and Cardinality work of Kindergarten.

Operations & Algebraic Thinking

Operations and Algebraic Thinking (OA) is a central domain within each course of IM K–5 v.360.

K–2: In Kindergarten, students begin exploring the OA domain through addition and subtraction in Unit 4, which supports their understanding of composing and decomposing numbers up to 10 in Unit 5. In Grade 1, students build on their understanding by adding and subtracting within 20. Finally, in Grade 2, students explore adding and subtracting within 100. The grade band culminates with students applying their understanding of this domain in Grade 2 Unit 8.

3–5: In Grade 3, students are introduced to multiplication and relating multiplication to division. Students apply this work in Grade 4 by investigating factors and multiples, solving word problems, and exploring multiplicative comparison and measurement. In Grade 5, students use the OA domain in various contexts to find volume, examine place value patterns, and apply operations to decimals.

Numbers & Operations in Base Ten

The Numbers and Operations in Base Ten (NBT) domain focuses on place value and operations ranging from multi-digit whole numbers to decimals up to the thousandths place.

K–2: The NBT domain is largely addressed in most units of Grades 1 and 2. In Kindergarten, students develop a strong understanding of numbers 0–20, which prepares them to work with numbers up to 99 in Grade 1. Here, students build familiarity with the base-ten system before adding within 100. Grade 2 furthers this understanding with a focus on adding and subtracting within 100, and continues moving students into numbers up to 1,000. Students finish Grade 2 with skills to add and subtract within 1,000.

3–5: In Grade 3, students start by working with whole numbers and addition and subtraction within 1,000. In Unit 4, they relate multiplication to division and use their knowledge of the place value system to multiply by multiples of 10. This work continues in Grade 4 as students apply the number system to larger multi-digit numbers in Unit 4 and multiply and divide multi-digit numbers in Unit 6. Grade 5 expands on and concludes these ideas in Units 4 and 5 where students explore place value patterns and decimal operations.

Numbers and Operations—Fractions

The Number and Operations—Fractions (NF) domain only applies to Grades 3–5, but it builds on the work in earlier grades in other domains.

3–5: The NF domain is first emphasized in Grade 3 when students begin to develop an understanding of fractions within the number system in Unit 5. This understanding continues to develop in Grade 4 as they compare fractions in Unit 2. In the next unit, students begin to multiply with fractions and relate fractions to decimals. In Grade 5, they add, subtract, multiply, and divide with fractions starting in Unit 2. This work prepares them to engage with the later Grade 5 concepts, multiplying and dividing fractions and more decimal and fraction operations.

Measurement and Data

In the Measurement and Data (MD) domain, students represent and interpret data, solve problems involving measurement, and work toward understanding concepts of perimeter, area, angle measures, and volume.

K–2: The MD domain is emphasized largely in Grades 1 and 2, while Kindergarten provides the foundational skills that students need in order to successfully develop key concepts. In Kindergarten, students work with classifying and comparing measurable attributes, including length. As they explore this domain in Grade 1, students learn concepts such as length measurements within 120 units and adding, subtracting, and working with data. In Grade 2, students build on that understanding by measuring length, practicing addition and subtraction on the number line, and adding, subtracting, and working with data.

3–5: Grade 3 lays a foundation for understanding geometric measurement with major concepts such as area and multiplication and two-dimensional shapes and perimeter. Grade 4 builds on this domain by exploring angles and angle measurement, and Grade 5 applies this work in Unit 1 as students are asked to find volume. In a similar trajectory, Grade 3 introduces students to different types of measurement, including length, time, liquid volume, and weight in Unit 6. Grade 4 builds on that work with multiplicative comparisons in Unit 5, and Grade 5 introduces conversions of these measures in Unit 6.

Geometry

In the Geometry (G) domain, students work with lines, angles, and shapes. They partition, examine attributes, and classify shapes based on their properties.

K–2: Each course in this grade band devotes an entire unit to the Geometry domain where the bulk of the work is concentrated. In Kindergarten, students specifically explore flat and solid shapes throughout two units. Grade 1 continues to examine shapes with the addition of time. Finally, Grade 2 builds on the previous work in the grade band with a special emphasis on geometry, time, and money.

3–5: Starting in Grade 3, students classify two-dimensional shapes based on their properties and also calculate perimeter. Grade 4 further examines shapes by studying angles in Unit 7 and exploring the properties of two-dimensional shapes in Unit 8. In Grade 5, students begin building the foundation for future work by working with the first quadrant in the coordinate plane in Unit 7.

Big Ideas by Grade Level

Each description of major concepts by grade level contains two tables to demonstrate how the units in IM v.360 map to California’s Big Ideas. The first table is organized by Big Idea and lists each unit that addresses its content. The second table is organized by unit title and lists each Big Idea that it addresses. These two tables share the same information in different formats to demonstrate the close alignment of IM v.360 and California’s Big Ideas. Each unit addresses at least one Big Idea, and each course covers all Big Ideas for the grade level.

In addition to California’s Big Ideas, the second table also showcases the California Common Core State Standards and Standards for Mathematical Practice that are central to each unit. The table lists standards that are addressed during that unit, though there may be additional standards that it builds on or builds toward. Similarly, while students have the opportunity to use all of the Standards for Mathematical Practice throughout each unit, the table highlights those that students are most likely to use. More details on content standards and Standards for Mathematical Practice can be found in the teacher materials for each lesson.

Each major concepts resource contains two exemplar lessons or activities that demonstrate how the curriculum directly supports other aspects of the California Framework. Please note that while language such as Content Connections and Drivers of Investigation may not be used to describe the examples, the lessons and activities have been intentionally chosen to address them across all courses.

The major concepts resources are intended to be viewed alongside the curriculum. Examples are referenced by their title and location in the curriculum rather than including the full text of every activity.

To get the most from the materials:

1. Navigate to the activity.
2. Skim the contents of the lesson it is contained within.
3. Read the full activity.
4. Read the description detailing the connection between this activity and the rest of the curriculum.

IM v.360 Grade 4 focuses on developing:

• understanding of and fluency with multi-digit multiplication
• understanding of dividing to find quotients involving multi-digit dividends
• understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers
• understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry

The diagram below links the major concepts in IM v.360 Grade 4 to the domains found in the California Common Core State Standards for Mathematics (CA CCSSM). Each shape represents a domain that is addressed within the major concept. The larger shapes represent the primary domain while the smaller shapes on the periphery represent secondary domains that are addressed. The arrows demonstrate how the major concepts are interconnected and build on each other.

Exemplars:

Unit 3: Extending Operations to Fractions Lesson 6: Problems with Equal Groups of Fractions AccessIM Link

Mathematical Practices: MP2, MP3, MP7 CA CCSSM Standards: 4.NF.4.a, 4.NF.4.b, 4.NF.4.c

Summary: In this lesson, students multiply whole numbers and fractions and match expressions to real-world situations as they solve problems within the context of a bakery. Students may choose to draw diagrams, write equations, or make use of patterns to understand the situations and answer the questions. As students multiply with unit fractions, they build on previous concepts involving operations with whole numbers and build towards future concepts involving decimals.

Unit 4: From Hundredths to Hundred-Thousands Lesson 17: Apply Rounding AccessIM Link	Mathematical Practices: MP2, MP3, MP4 CA CCSSM Standard: 4.NBT.3
Summary: In this lesson, students work within the real-world context of airplane altitudes to round multi-digit numbers. As they practice rounding, they analyze the impact of rounding to different place values to help them make decisions. This work builds on previous work with place value and builds toward operations with multi-digit numbers.

California’s Big Ideas Within the IM Course

Measuring and Plotting	Rectangle Investigations	Number and Shape Patterns	Factors and Area Models	Multi-Digit Numbers	Fraction Flexibility	Visual Fraction Models	Circles, Fractions, and Decimals	Shapes and Symmetries	Connected Problem Solving
Units 2–3 Units 5–6 Unit 8	Units 5–8	Unit 1 Units 4–7	Unit 1 Units 5–8	Units 4–8	Units 2–6 Unit 8	Units 2–4	Units 3–8	Unit 4 Units 7–8	Unit 1 Units 4–8

IM v.360 Grade 4 Major Concepts	CA CCSSM Domains*	California’s Big Ideas	CA Standards	MPs
Unit 1: Factors and Multiples	OA	Number and shape patterns Factors and area models Connected problem solving	OA.3 OA.4 OA.5	MP1 MP4
Unit 2: Fraction Equivalence and Comparison	NF	Measuring and plotting Fraction flexibility Visual fraction models	NF.1 NF.2	MP3 MP6 MP7 MP8
Unit 3: Extending Operations to Fractions	NF MD	Measuring and plotting Fraction flexibility Visual fraction models Circles, fractions, and decimals	MD.4 NF.1 NF.2 NF.3 NF.4 NF.5	MP1 MP2 MP3 MP6 MP7 MP8
Unit 4: From Hundredths to Hundred-Thousands	NBT NF	Number and shape patterns Multi-digit numbers Fraction flexibility Visual fraction models Circles, fractions, and decimals Shapes and symmetries Connected problem solving	NBT.1 NBT.2 NBT.3 NBT.4 NF.3.c NF.5 NF.6 NF.7	MP2 MP3 MP6 MP7
Unit 5: Multiplicative Comparison and Measurement	OA NBT NF MD	Measuring and plotting Rectangle investigations Number and shape patterns Factors and area models Multi-digit numbers Fraction flexibility Circles, fractions, and decimals Connected problem solving	MD.1 MD.2 MD.3 NBT.4 NBT.5 NBT.6 NF.4 OA.1 OA.2 OA.3	MP2 MP3 MP6 MP7
Unit 6: Multiplying and Dividing Multi-digit Numbers	OA NBT MD	Measuring and plotting Rectangle investigations Number and shape patterns Factors and area models Multi-digit numbers Fraction flexibility Circles, fractions, and decimals Connected problem solving	MD.1 MD.2 MD.3 NBT.4 NBT.5 NBT.6 OA.1 OA.2 OA.3 OA.4 OA.5	MP2 MP3 MP7
Unit 7: Angles and Angle Measurement	NBT MD G	Rectangle investigations Number and shape patterns Factors and area models Multi-digit numbers Circles, fractions, and decimals Shapes and symmetries Connected problem solving	G.1 MD.5 MD.6 MD.7 NBT.4 NBT.5 NBT.6	MP6 MP7
Unit 8: Properties of Two- dimensional Shapes	NF MD G	Measuring and plotting Rectangle investigations Factors and area models Multi-digit numbers Fraction flexibility Circles, fractions, and decimals Shapes and symmetries Connected problem solving	G.1 G.2 G.3 MD.1 MD.2 MD.3 MD.5 MD.6 MD.7 NBT.4 NBT.5 NBT.6 NF.3.c NF.4	MP5 MP6 MP7

*bolded text indicates primary domain