Developing Place Value Understanding and Additive Reasoning
To build a true conceptual understanding of numbers, students need a variety of experiences in building,
representing, and decomposing numbers. When students understand how numbers are structured in the
base-ten system, they will reason flexibly and apply efficient strategies to addition and subtraction
problems. On this collections page, we show a progression of experiences that allow students to build
symbolic reasoning rooted in concrete and semi-concrete experiences.
Developing Place-Value Reasoning with Manipulatives
To develop conceptual place value reasoning, students need experience building, representing, comparing, and decomposing multi-digit numbers using a variety of place-value manipulatives.
Place-Value Manipulatives
Learn more about a variety of place value manipulatives and how groupable and pre-grouped materials can support students as they make sense of quantity and structure.
KY.1.NBT.2, KY.2.NBT.1- Building Two-Digit Numbers with Bundles and Sticks
Explore building and representing numbers using bundles and sticks, then consider what happens when a stick or bundle is added or removed, bundling or unbundling a 10 as needed.
KY.1.NBT.1, KY.1.NBT.2, KY.1.NBT.5 - Introducing the 100 Bead String
Learn more about the 100-bead string as a tool to show the tens and ones structure of numbers in a linear manner, and how it allows students to build and compare two-digit numbers.
KY.K.NBT.1, KY.1.NBT.2, KY.1.NBT.3 - Flexibly Decomposing Numbers into Base-Ten Units
Building and representing numbers in different ways with base ten materials provides a foundation for students to operate flexibly.
KY.2.NBT.3, KY.3.NBT.2, KY.4.NBT.2 
Counting Large Collections (Grades 3-6)
Learn more about facilitating the counting collections routine with larger numbers. When counting large collections (over 1000), students will find effective ways to group and organize items into manageable chunks, helping them to develop a deeper understanding of base-ten units and coordinate between place-value units.
KY.3.NBT.1, KY.4.NBT.1, KY.4.NBT.2
Developing Foundations for Fluency with Multi-digit Addition & Subtraction
To build fluency, students need to develop a deep understanding of the structure of numbers, including how numbers can be composed and decomposed, particularly around tens. Working with addition and subtraction problems within 20, along with two-digit with one-digit addition and subtraction problems, will allow students to build a bridge between basic fact fluency reasoning and fluent addition and subtraction of larger numbers.
Developing Fluency with Addition & Subtraction within 20
Throughout first and second grades, students should develop a flexible repertoire of strategies utilizing structure and relationships between numbers when adding and subtracting within 20.
- Strategy Instruction: Addition
Learn more about the Addition Fact Fluency Learning Progression from Jennifer Bay-Williams, author of Math Fact Fluency.
KY.K.OA.4, KY.1.OA.6, KY.2.OA.2 - Single-Digit Addition: A Range of Strategies for Solving 8 + 7
Walk through various strategies students might use to solve 8 + 7, including counting on, make-a-ten, pretend-a-ten, and near doubles.
KY.1.OA.6, KY.2.OA.2 - Subtraction within 20: A Range of Strategies for Solving 14-8
Understand that subtraction can be thought of as either take-away or distance apart, and then explore a range of strategies for 14-8, including counting back in chunks, compensation, relating to a double, and think addition.
KY.1.OA.3, KY.1.OA.4, KY.1.OA.5, KY.1.OA.6, KY.2.OA.2
Early Addition and Subtraction Strategies within 100 (2-digit +/- 1-digit)
Students solve two-digit with one-digit problems by relying on the structure of numbers, such as making tens, composing and decomposing numbers, and bridging to benchmarks.
- Early Addition Strategies within 100: Mini Ten Frames
Explore a progression of prompts using ten frames that support students in developing non-count-by-ones strategies when adding a 2-digit number plus a 1-digit number.
KY.1.NBT.4 - Early Addition Strategies within 100: Bead String
Explore a progression of addition prompts presented on a 100 bead string that help students see 2-digit numbers in relation to tens.
KY.1.NBT.1, KY.1.NBT.4, KY.2.NBT.5 - Early Subtraction Strategies within 100: Bead String
Explore a progression of subtraction prompts presented on a 100 bead string that help students develop early subtraction strategies using multiples of 10 as a benchmark.
KY.1.NBT.1, KY.2.NBT.5
Addition & Subtraction Strategies within 100
Students should use a range of strategies when solving two-digit addition and subtraction problems. Students should be developing their understanding of addition and subtraction simultaneously, understand that addition and subtraction are inverse operations, and look for connections between addition and subtraction strategies. This section highlights how teachers and students might use base-ten manipulatives, symbolic representations, visual annotations, and number talk routines to reason flexibly and efficiently.
Addition Strategies
- Two-Digit Addition: Counting On and Partial Sums
Learn about how students might use counting on by tens and ones or partial sums to add two 2-digit numbers.
KY.2.NBT.5 - Addition within 100: Make Tens Strategy
Learn how students might use the make tens strategy for addition, starting with one-digit addends and extending into two-digit addends, modeled using a variety of materials, including ten frames, 2-row Rekenreks, ten-frame cards, unifix cube towers, and bundles and sticks.
KY.2.OA.2, KY.2.NBT.5 - Addition within 100: Compensation Strategy
Learn more about how students might use the compensation strategy to add 1- or 2-digit numbers.
KY.2.OA.2, KY.2.NBT.5 - Addition: A Range of Strategies for Solving 79 + 48
Explore a range of strategies including counting on in chunks, using benchmark numbers, making tens, partial sums, and compensation, and includes how a teacher might record students' thinking on a number line and with symbolic notation.
KY.2.NBT.5, KY.3.NBT.2 
Problem Strings: Using Strategies to Add within 100
Watch second graders share and name their strategies to solve 8 + 5, 28 + 5, and 28 + 15 as part of a problem string their teacher carefully crafted to help students see connections and relationships, promoting the discovery and practice of new strategies.
KY.2.NBT.5
Subtraction Strategies
- Two-Digit Subtraction: Counting Back and Partial Differences
Learn more about how students might use counting back in chunks and partial differences to solve subtraction tasks within 100.
KY.2.NBT.5 - Two-Digit Subtraction: Compensation Strategy
Learn about how students might use the compensation strategy to subtract 2-digit numbers using concrete, semi-concrete, and abstract representations.
KY.2.NBT.5 - Subtraction: A Range of Strategies for Solving 53 - 29
Explore a range of strategies including counting back in chunks, using benchmark numbers, partial differences, and think-addition, and includes how a teacher might record students' thinking on a number line and with symbolic notation.
KY.2.NBT.5, KY.3.NBT.2
Extending Strategies to larger whole number, decimals, and fractions
Mathematical reasoning involves recognizing the underlying structures and relationship within our number system. This section illustrates how the same foundational strategies students learned in early grades such as "Make Tens" and the "Same Difference" strategies remain powerful and relevant as they progress. By extending these methods to multi-digit whole numbers, decimals, and fractions, students see the core principles of additive reasoning stay consistent, regardless of how large or complex the numbers become.
- Addition: The Make Tens Strategy
Learn how the make tens strategy for addition can be extended to multi-digit whole numbers, decimals, and even fractions.
KY.2.NBT.5, KY.3.NBT.2, KY.4.NF.3, KY.5.NBT.7 - Subtraction: The Same Difference Strategy
Learn how the same difference strategy for subtraction can be extended to multi-digit whole numbers, decimals, and even fractions.
KY.2.NBT.5, KY.3.NBT.2, KY.4.NF.3, KY.5.NBT.7