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.