The big ideas that define units in InsightMath provide a framework for making connections across the domains and clusters defined by the Common Core State Standards for Mathematics. A description of where and how domains and clusters are connected within each big idea is described in the table below.
Connections Across Domains and Clusters - Grade 6
| Unit | Big Idea | Connections Across Domains and Clusters | Major Clusters |
Additional/ Supporting Clusters |
Unit 1Discovering Algebraic Equations |
Algebra is a way of expressing generalizations.
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Students’ schema for representing numbers and operations expands as they formalize their understanding of algebraic expressions. Building on their experience using letters to represent specific unknown values in equations from grades 3–5, students develop a deeper understanding that variables can represent any number in a specified set (6.EE.B) and use them to express generalizations. Throughout the unit, students connect their knowledge of operations with numbers to working with variables, noticing that the same properties of operations apply in algebra as they do in arithmetic (6.EE.A). Students transform their schema of expressions, recognizing that algebra is a language for expressing not just specific values but patterns, relationships, and generalizations that can be applied to many situations.
|
6.EE.A 6.EE.B
|
None |
Unit 2Exploring Equivalence |
Any number or expression can be represented in an infinite number of ways that have the same value.
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Students enter grade 6 with schemas for recognizing that numerical expressions can be written in different, equivalent ways. In this unit, students deepen their understanding, exploring common factors and multiples (6.NS.B) with fractions and numerical expressions, and then applying the same thinking to more complex expressions with variables. Students apply their existing schemas for operations and their properties to develop a more sophisticated understanding of equivalence, learning to systematically identify, verify, and generate equivalent expressions (6.EE.A).
|
6.EE.A | 6.NS.B |
Unit 3Discovering Ratios |
Numbers and measures can be compared by their relative values.
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In this unit, students expand their schema of multiplicative comparison as they develop an understanding of ratios to compare quantities and express proportional relationships. Students extend their understanding beyond “how many more” and “how many times as many” to consider how a proportional relationship can be described with a ratio (6.RP.A). Students apply their understanding of equivalence and their skills in working with common factors and multiples (6.NS.B) from the previous unit to recognize and generate equivalent ratios. Students also examine proportional relationships on the coordinate plane (6.NS.C), noticing patterns when one variable is graphed against the other and introducing algebraic and graphic approaches to solving the same problem.
|
6.RP.A 6.NS.C |
6.NS.B |
Unit 4Discovering Rates, Percentages, and Proportional Data |
Rates and percentages are specific ratios that help compare numbers or measures by their relative values.
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In this unit, students expand their understanding of ratios to include two specialized forms: rates and percentages (6.RP.A). Building on their prior work with equivalent ratios, students develop a schema for thinking about the relationship between quantities by comparing them to 1 unit or to 100 units. Students develop a schema for recognizing independent and dependent variables in real-world problems and representing proportional relationships with equations (6.EE.C). Their schema about variables grows as they explore how the value of one variable in a relationship impacts the value of the other (6.EE.B). Students also continue to explore the graphs of proportional relationships (6.NS.C), recognizing how to determine rate either algebraically or graphically and laying the foundation for a formal understanding of slope in future grade levels. They also build their schema for data analysis, using proportional rather than absolute reasoning to make comparisons. |
6.RP.A 6.NS.C 6.EE.B 6.EE.C |
None |
Unit 5Solving Algebraic Equations |
Algebraic equations can be used to model and solve real-world problems.
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In this unit, students further build their schema of equivalence by developing their understanding of equations as relationships that can be manipulated while maintaining equivalence (6.EE.A), connecting concrete models to abstract representations. They also build their schema of variables, recognizing that equations can be true for all values of the variable or only particular values (6.EE.B). Students rely on notions of equivalence and inverse operations to solve for an unknown value, their first experience with formal algebraic manipulation to solve equations (6.EE.B). They apply these skills to real-world situations, representing them algebraically and graphically (6.EE.C).
|
6.EE.A 6.EE.B 6.EE.C |
None |
Unit 6Extending Fraction Operations |
Division can be performed by multiplying due to the inverse relationship between multiplication and division.
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In this unit, students expand their fraction operation schema to understand division of fractions by fractions through concrete models and abstract representations. Students’ mental frameworks for division are challenged as they recognize that dividing by a fraction is equivalent to multiplying by its reciprocal (6.NS.A). They discover how using common factors to simplify within and across fractions (6.NS.B) can help make computation with fractions easier. Students make connections to their learning earlier in the year by writing and solving equations involving percentages written as fractions in real-world situations (6.RP.A and 6.EE.B). This unit transforms students’ schema of algebraic thinking as they connect division with multiplication, recognize patterns in quotients, and apply these relationships to solve complex equation forms and real-world problems involving area and measurement.
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6.RP.A 6.NS.A 6.EE.B |
6.NS.B |
Unit 7Extending Arithmetic in Base Ten |
The base-ten system allows all four operations to be performed with algorithms involving a series of single-digit computations.
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Students enter grade 6 with a foundational understanding of place value and the four operations with whole numbers and decimals. In this unit, students deepen their procedural fluency with addition, subtraction, multiplication, and division to include multidigit decimal numbers (6.NS.B). They refine their understanding of how the standard algorithms leverage place value patterns and properties of operations to perform calculations efficiently, recognizing that all four operations can be performed through a series of single-digit computations regardless of the size or type of number. This unit gives students opportunities to interpret situations and compute using the appropriate operation in a variety of situations, including solving equations involving percentages written as decimals (6.EE.B and 6.RP.A).
|
6.RP.A 6.EE.B
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6.NS.B |
Unit 8Extending the Number System |
The number system can be extended to include negative numbers which are reflections of their positive counterparts over the origin.
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In this unit, students expand their number schema to include negative numbers, transforming their understanding of the number system from a linear progression starting at 0 to a bidirectional system extending infinitely in both directions (6.NS.C). Students develop mathematical symmetry as a powerful concept, seeing how the number line reflects across 0, with each positive number having an opposite negative counterpart at the same distance from the origin. This expanded number schema allows students to quantify and reason about quantities having opposite directions or values—such as elevations above and below sea level, temperatures above and below freezing, and financial positions of credit and debt—enriching their ability to model and analyze real-world contexts mathematically (6.NS.C) and to represent solutions to inequalities on the number line (6.EE.B).
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6.NS.C 6.EE.B |
None |
Unit 9Exploring Statistics |
How data is represented and analyzed can impact how it is interpreted.
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In this unit, students develop their schema for statistical thinking by systematically investigating data distributions. Their understanding of measures of center and variability evolves as they explore how different representations highlight various aspects of datasets (6.SP.A and 6.SP.B). Students develop a deeper schema for statistical reasoning by connecting visual representations, numerical summaries, and contextual interpretations, recognizing how the choice of data display and analysis method impacts the stories data can tell (6.SP.B).
|
None |
6.SP.A 6.SP.B |
Unit 10Exploring Two- and Three-Dimensional Space |
The size of objects can be quantified in one, two, or three dimensions to serve a particular purpose or context.
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Students expand their understanding of how size can be quantified in one, two, or three dimensions as they develop their measurement schema. In this unit, students recognize that the same object can be measured in multiple ways—by its perimeter (one dimension), area (two dimensions), surface area, or volume (three dimensions)—with each measurement serving a different purpose and requiring appropriate units (6.G.A). They derive formulas for two-dimensional shapes and use the coordinate plane as a tool to determine lengths (6.NS.C). Students connect their existing knowledge of area and perimeter to new understandings of surface area and volume, recognizing how the dimensional power of the measurement units (linear, square, cubic) corresponds directly to the dimensions being measured (6.EE.A).
|
6.EE.A 6.NS.C |
6.G.A |