Addition & Subtraction of Fractions - Explore why fractions need compatible units before their quantities can be meaningfully combined or compared.

Addition & Subtraction Story Problems - Learn how different story structures represent addition and subtraction and influence the strategies used to solve them.

Algebraic Equality - Explore equality as a relationship between two expressions that have the same value rather than as a signal to perform a calculation.

Algebraic Equations - Interpret equations as statements about equal quantities and use reasoning to determine values that make those statements true.

Algebraic Expressions - Use variables and operations to represent unknown values, changing quantities, patterns, and relationships.

Analyzing Addition & Subtraction Strategies - Compare different solution methods to understand their reasoning, efficiency, and mathematical connections.

Angle Measurement - Understand angles as measures of rotation and use degrees, benchmarks, and tools such as protractors to describe their size.

Attributes - Identify and describe the measurable and defining characteristics that can be used to compare and classify geometric figures.

Balance Models for Equations - Use balance models to visualize equality and explain why the same mathematical operation must preserve both sides of an equation.

Bar Charts & Pie Charts - Use bar charts and pie charts to represent categorical data and compare when each display is most useful.

Basic Probability - Understand probability as a measure of how likely an event is to occur and represent likelihood using numbers from 0 to 1.

Benchmark Percents & Estimation - Use familiar percents such as 1%, 10%, 25%, 50%, and 100% to estimate and reason efficiently about unfamiliar percentages.

Bias in Data Collection - Identify ways that sampling, question wording, measurement, and other collection methods can systematically distort data.

Building Mental Addition Strategies - Use number relationships and place value to develop flexible strategies for adding mentally.

Building Mental Subtraction Strategies - Use number relationships, decomposition, and place value to reason efficiently about subtraction.

Building Numbers in Base Ten - Learn how numbers can be built, broken apart, and regrouped using ones, tens, hundreds, and other powers of ten.

Center, Shape, Spread, & Outliers - Describe a distribution as a whole by examining its typical value, overall form, variability, and any unusually distant observations.

Choosing an Appropriate Data Display - Select a graph or display that matches the type of data and the question being investigated.

Circles - Understand circles through radius, diameter, circumference, area, and the defining idea of equal distance from a center.

Comparing Alternative Algorithms - Compare multiple multiplication and division algorithms to identify the mathematical ideas they share and the ways they differ.

Comparing Distributions - Compare two or more data sets by considering differences in their centers, spreads, shapes, and unusual values.

Comparing Linear & Nonlinear Patterns - Use tables, graphs, equations, and patterns to distinguish linear relationships from nonlinear ones.

Congruency & Similarity - Compare figures to determine whether they have exactly the same size and shape or the same shape at different sizes.

Connecting Area, Surface Area, & Volume - Compare linear, square, and cubic measurement to understand how area, surface area, and volume are related but measure different attributes.

Connecting Fractions, Decimals, Ratios, & Percents - Recognize fractions, decimals, ratios, and percents as interconnected ways of representing the same quantitative relationships.

Connections Between Multiplication & Division - Examine how multiplication and division are inverse operations and how each can be used to understand the other.

Constant vs. Changing Rate of Change - Determine whether a relationship changes at a constant rate or whether its rate itself varies.

Conversions - Use relationships among units to convert measurements while reasoning about why numerical values change when units change.

Counting Strategies for Addition & Subtraction - Develop counting-based strategies that bridge direct modeling and more efficient numerical reasoning.

Defining Ratios - Understand ratios as comparisons between quantities and explore the different ways ratio relationships can be represented and interpreted.

Describing Geometry - Use precise mathematical language, properties, and relationships to communicate geometric reasoning clearly.

Describing Geometry Using Precise Mathematical Language - Practice using definitions, attributes, and geometric vocabulary to describe shapes and relationships clearly and accurately.

Direct Modeling with Objects & Drawings - Use physical objects and visual representations to model addition and subtraction situations directly.

Displaying Categorical Data - Organize and represent data made up of groups or categories so that frequencies and comparisons can be seen clearly.

Displaying Numerical Data - Represent quantitative data in ways that reveal patterns in values, frequencies, and distributions.

Divisibility & Divisibility Rules - Use place-value patterns and number properties to determine whether one number divides evenly into another.

Division Algorithms - Examine how written division methods organize repeated grouping, subtraction, and place-value reasoning.

Division of Fractions - Develop meaning for fraction division by reasoning about sharing, measurement, and the number of groups in a quantity.

Division Strategies - Use models, known facts, and number relationships to solve division problems efficiently.

Equivalent Fractions - Explore how different fraction names can represent the same quantity and why equivalence works.

Experimental vs. Theoretical Probability - Compare probabilities predicted by mathematical models with probabilities observed through actual or simulated experiments.

Experiments & Simulations - Use repeated trials and simulations to model random processes and investigate long-run patterns.

Explaining Mathematical Reasoning - Communicate not only an answer but also the mathematical evidence and reasoning that show why the answer makes sense.

Extending Place Value - Extend the place-value system in both directions to understand decimals and the relationships created by multiplying and dividing by powers of ten.

Finding Rules in Repetition - Analyze repeating patterns to identify the underlying rule that determines how the pattern continues.

Fraction Comparisons - Compare and order fractions using benchmarks, common units, visual models, and reasoning about size.

Factors & Prime Factorizations - Break whole numbers into factors and represent them uniquely as products of prime numbers.

Fractions as Part-Whole, Measure, Quotient, & Operator - Examine several meanings of fractions and how each interpretation appears in different mathematical situations.

Fractions on the Number Line - Locate fractions on the number line to emphasize that fractions are numbers with specific magnitudes.

Fractions to Decimals - Connect fraction and decimal notation as different ways of representing the same numerical quantities.

Function Tables - Organize input-output relationships in tables and use patterns in the values to understand or identify a function rule.

Generalizing a Pattern with Variables - Use variables and expressions to describe a pattern in a way that applies to any stage or term.

Geometric Patterns - Explore patterns created with shapes and visual arrangements and describe how their structure changes from one stage to the next.

Geometric Problems - Apply geometric ideas and relationships to solve problems that require visualization, reasoning, and connecting multiple concepts.

Graphing Functions - Represent functions on coordinate graphs to visualize how one quantity changes in relation to another.

Greatest Common Factors - Identify the largest factor shared by two or more numbers and connect it to their underlying factor structure.

Hierarchical Classification of Shapes - Organize shapes into nested categories to understand how one figure can belong to several geometric classes at the same time.

Histograms & Dotplots - Use histograms and dotplots to examine the shape, concentration, gaps, and spread of numerical data.

How Counting Actually Works - Discover why counting is more than simply saying numbers in order. Explore the principles that make counting reliable and meaningful.

How Big Is It Really? - Develop a sense of how large or small numbers are by comparing quantities, estimating values, and reasoning about numerical magnitude.

Inputs, Outputs, & Rules - Understand functions as rules that connect allowable input values to corresponding output values.

Interior & Exterior Angles - Investigate relationships among angles inside and outside polygons and use those relationships to determine unknown measures.

Interpreting Misleading Graphs - Identify design choices and distortions that can make a graph give a misleading impression of the underlying data.

Introduction to Fractions - Develop the idea of a fraction as a number representing a quantity made from equal-sized parts.

Least Common Multiples - Find the smallest multiple shared by two or more numbers and explore how common multiples arise from number structure.

Linear Functions - Explore relationships with a constant rate of change and connect their tables, graphs, equations, and real-world meanings.

Lines & Angles - Explore how lines, rays, segments, and intersecting lines create important angle relationships.

Looking for Structure - Identify useful patterns, properties, and relationships within a problem that can make unfamiliar mathematics easier to understand.

Making and Testing Conjectures - Form mathematical predictions from observed patterns and test them using examples, counterexamples, and logical reasoning.

Making Friendly Numbers - Adjust numbers strategically to create easier calculations while preserving the value of an expression.

Making Sense of Division - Interpret division through sharing, grouping, and determining how many groups or how much is in each group.

Meaning of the Equal Sign - Examine the equal sign as a symbol expressing equivalence between two quantities or expressions on either side.

Measures of Center - Use mean, median, and other measures of center to describe a typical or representative value in a data set.

Measures of Spread - Compare ways of describing variability, including range, interquartile range, and standard deviation.

Multiple Representations of Functions - Connect verbal descriptions, rules, tables, graphs, and other representations of the same functional relationship.

Multiplication Algorithms - Explore how written multiplication methods organize place-value reasoning to calculate products efficiently.

Multiplication & Division Story Problems - Explore the different problem structures that give multiplication and division their mathematical meaning.

Multiplication of Fractions - Interpret fraction multiplication through scaling, area models, and finding a fractional part of a quantity.

Multiplication Strategies - Develop flexible multiplication strategies using equal groups, arrays, decomposition, and known facts.

Nets of Three-Dimensional Figures - Explore how two-dimensional arrangements of faces can fold to form three-dimensional objects.

Number Patterns - Identify relationships within sequences of numbers and use those relationships to predict and extend patterns.

Numerical Expressions - Interpret, construct, and evaluate expressions involving numbers and operations while paying attention to their mathematical structure.

Other Functions - Examine examples of nonlinear relationships and compare their behavior with familiar linear patterns.

Partial Products & Partial Quotients - Use place value to break multiplication and division into manageable partial calculations.

Percent Increase & Decrease - Analyze how quantities change by a given percent and distinguish the original amount, amount of change, and resulting value.

Perimeter & Area - Compare the distance around a figure with the amount of surface it covers and explore how these measurements can vary independently.

Polygons - Identify and classify polygons using defining attributes such as straight sides, closure, regularity, and convexity.

Polyhedra - Identify and classify three-dimensional figures by examining their faces, edges, vertices, and other defining properties.

Population vs. Sample - Distinguish the full group of interest from the smaller group actually observed and examine how samples are used to learn about populations.

Prime vs. Composite Numbers - Distinguish prime and composite numbers by examining the number and structure of their factors.

Probability Rules - Use fundamental probability rules to determine the likelihood of individual, combined, and complementary events.

Properties of Operations as Foundations for Algebra - Connect the commutative, associative, and distributive properties to equivalent expressions and the reasoning used throughout algebra.

Properties vs. Definitions - Distinguish characteristics that are always true of a shape from the minimum attributes needed to define that shape.

Proportional vs. Nonproportional Relationships - Distinguish proportional relationships from other relationships by examining how quantities change relative to one another.

Quadrilaterals - Explore the properties and relationships among four-sided figures, including parallelograms, rectangles, rhombi, squares, and trapezoids.

Quartiles & Boxplots - Use quartiles and boxplots to summarize the center and spread of data and identify how values are distributed across four sections.

Rates & Unit Rates - Compare quantities measured in different units and use unit rates to describe how much of one quantity corresponds to one unit of another.

Reasoning About Percent - Understand percent as a comparison to 100 and use part-whole relationships to solve a variety of percent problems.

Recursive vs. Explicit Patterns - Compare rules that describe a term using previous terms with rules that determine any term directly from its position.

Relational Thinking - Use relationships among numbers and expressions to reason about equality without necessarily carrying out every calculation.

Representing Numbers in Different Ways - Learn how the same number can be represented using objects, pictures, words, symbols, and expanded forms - and how moving between representations strengthens number sense.

Sample Spaces & Outcomes - Identify all possible outcomes of a chance process and organize them into a sample space for probability analysis.

Sampling Methods - Compare different methods for selecting samples and consider how the method used can affect the quality of the resulting data.

Scale Factor - Use multiplicative relationships to describe how corresponding lengths change when figures are enlarged or reduced.Solving Ratios

Solving Ratios - Use scaling, equivalent ratios, and visual models to determine unknown quantities while preserving a proportional relationship.

Standard Deviation - Measure how far data values typically lie from the mean and use standard deviation to describe overall variability.

Statistical Questions - Identify questions that anticipate variability in their answers and can be investigated through the collection and analysis of data.

Surface Area - Measure the total area covering the outside of a three-dimensional figure by connecting its faces to two-dimensional area.

Tables, Graphs, & Double Number Lines for Ratios - Represent ratio relationships in multiple ways and connect the same proportional structure across tables, graphs, and double number lines.

Transformations & Symmetry - Explore ways figures can move or map onto themselves while identifying patterns of reflectional and rotational symmetry.

Translations, Rotations, & Reflections - Examine the three rigid transformations and describe how each changes a figure’s position or orientation without changing its size or shape.

Triangles - Classify triangles by their sides and angles while examining the properties that all triangles share.

True, False, & Open Number Sentences - Determine whether mathematical statements are true or false and identify values that make open number sentences true.

Types of Data - Distinguish among different kinds of categorical and numerical data and consider how data type influences analysis.

Understanding Quantity - Explore how counting connects numbers to actual quantities and why matching one number to one object is fundamental to understanding number.

Understanding the Standard Addition Algorithm - Examine how the standard addition algorithm uses place value to organize and combine quantities efficiently.

Understanding the Standard Subtraction Algorithm - Explore how the standard subtraction algorithm uses place value and decomposition to find differences.

Understanding Zero - Explore the many roles of zero: representing no quantity, holding a place in our number system, and acting as an important mathematical identity.

Units & Iteration - Understand measurement as repeatedly applying a chosen unit to determine the size of a length, area, or other quantity.

Variables as Unknowns, Changing Quantities, & Generalized Numbers - Explore the different roles variables can play and move beyond the idea that a letter always represents a single missing number.

Visual Models for Fraction Operations - Use number lines, area models, and other representations to explain why fraction operation procedures work.

Volume - Understand volume as the amount of three-dimensional space a solid occupies, measured using cubic units.

Weighted Averages - Calculate and interpret averages in which some values contribute more heavily than others.

What Does Addition Mean? - Explore addition as joining, combining, and increasing quantities in different mathematical situations.

What Does Subtraction Mean? - Examine subtraction as taking away, comparing, and finding an unknown part or difference.

What If We Counted Differently? - Explore number systems that use bases other than ten and discover which features of place value are universal and which depend on the base being used.

Which Number is Really Bigger? - Use place value to read, write, order, and compare numbers while developing strategies for determining their relative size.

Why Area Formulas Work - Use decomposition, rearrangement, and visual models to explain the reasoning behind common area formulas.

Why Digits Mean What They Mean - Explore why a digit's value depends on where it appears and how the structure of place value makes our number system work.

Why Regrouping Works - Connect regrouping procedures to the underlying place-value relationships among ones, tens, hundreds, and beyond.