Module 1: Counting & Understanding Quantity

Explore what counting actually means and how numbers communicate quantity. This module develops the ideas behind counting, cardinality, estimation, number representation, and zero.

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

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

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.

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.

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.

Module 2: Place Value & Number Representation

Investigate how the position of a digit determines its value and how numbers are built within a place-value system. Extend these ideas from whole numbers to decimals, powers of ten, and number systems beyond base ten.

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.

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

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

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.

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.

Module 3: Meaning of Addition & Subtraction

Build a conceptual understanding of addition and subtraction by exploring their meanings, problem types, models, and early solution strategies.

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.

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

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

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

Module 4: Addition & Subtraction Strategies & Algorithms

Develop flexible addition and subtraction strategies while examining how place value supports efficient mental methods and written algorithms.

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.

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

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.

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

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

Module 5: Multiplication & Division Concepts

Develop meaning for multiplication and division through problem situations, models, strategies, and connections between the two operations.

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

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

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

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

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

Module 6: Multiplication & Division Algorithms

Connect conceptual models and place-value reasoning to efficient written methods for multiplication and division.

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

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

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

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

Module 7: Fractions as Numbers

Build a strong understanding of fractions as numbers by examining their meanings, representations, equivalence, size, and connections to decimals.

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

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

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

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

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

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

Module 8: Fraction Operations

Develop conceptual meaning for fraction operations by connecting calculations to visual models, quantities, and number relationships.

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

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

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

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

Module 9: Factors & Number Structure

Investigate the multiplicative structure of whole numbers through factors, primes, divisibility, and shared numerical relationships.

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

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

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

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

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

Module 10: Angles, Lines, & Geometric Language

Build the language and reasoning needed to describe geometric relationships precisely, with a focus on angles, lines, attributes, and mathematical definitions.

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

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

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

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

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

Module 11: Two-Dimensional Shapes

Examine the properties and relationships of two-dimensional figures while developing a flexible, hierarchical understanding of geometric classification.

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

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

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

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

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

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

Module 12: Perimeter, Area, & Measurement

Develop measurement as a process of comparing quantities to units while exploring perimeter, area, unit conversions, and the reasoning behind familiar formulas.

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

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

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

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

Module 13: Three-Dimensional Geometry

Extend geometric and measurement reasoning into three dimensions by examining solids, nets, surface area, volume, and the relationships among different types of measurement.

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

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

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

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

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.

Module 14: Transformations, Congruence, & Similarity

Explore how figures can move, change size, and remain mathematically related through transformations, symmetry, congruence, similarity, and scale.

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

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

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.

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

Module 15: Ratios & Proportional Reasoning

Develop proportional reasoning by comparing quantities, examining rates, and representing relationships with tables, graphs, models, and equations.

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

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

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.

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

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.

Module 16: Percent Reasoning

Build a flexible understanding of percent as a proportional relationship and connect percents to fractions, decimals, estimation, and real-world changes.

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

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

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

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

Module 17: Numerical & Algebraic Expressions

Develop algebraic thinking by examining how numbers, variables, operations, and mathematical properties can be used to represent calculations, patterns, and general relationships.

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

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

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.

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

Module 18: Equality & Equations

Develop a relational understanding of equality and use that understanding to interpret, create, analyze, and solve equations.

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.

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

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

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

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

Module 19: Patterns & Generalizations

Investigate visual and numerical patterns, identify how they grow or repeat, and develop increasingly general ways to describe their structure.

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

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

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

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

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

Module 20: Introduction to Functions

Develop the idea of a function as a relationship between quantities and connect function rules to tables, graphs, and other representations.

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

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

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

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

Module 21: Linear & Nonlinear Relationships

Compare different types of functional relationships by examining patterns of change, graphical behavior, and the distinction between constant and changing rates.

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

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

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

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

Module 22: Mathematical Structure & Problem Solving

Strengthen mathematical problem solving by looking for structure, making conjectures, describing relationships precisely, and clearly explaining the reasoning behind a solution.

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

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

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.

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

Module 23: Foundations of Probability

Develop a conceptual understanding of probability by examining outcomes, sample spaces, experiments, simulations, and the relationship between theoretical and experimental results.

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

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

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

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

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

Module 24: Collecting & Understanding Data

Examine how statistical investigations begin by asking appropriate questions, identifying populations and samples, choosing sampling methods, and considering sources of bias.

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

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

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

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.

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

Module 25: Displaying Data

Represent categorical and numerical data using appropriate displays and develop the ability to choose, interpret, and critically evaluate graphical representations.

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

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

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

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

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

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

Module 26: Describing & Comparing Data

Describe distributions using measures of center and spread, graphical summaries, and overall features such as shape and outliers, then use those features to compare data sets meaningfully.

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

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

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

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

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

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

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