# Bridges Intervention Volumes

Bridges Intervention was written to address key numeracy and computation skills and concepts for each grade level, K–5. Specifically, Bridges Intervention deals with those clusters identified as major within the following domains: Counting & Cardinality (K), Operations & Algebraic Thinking (K–4), Number & Operations in Base Ten (K–5), and Number & Operations: Fractions (3–5).

## Number: Counting & Early Place Value

The eight modules in this volume focus on the skills and concepts needed to count, read, write, compare, and understand numbers to 5, 10, 20, 120, and 1,000. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one or more of the following:

- Counting forward and backward by 1s to 5, 10, 20, 100, 120, or 1,000 starting from any number within the range
- Counting up to 20 objects with one-to-one correspondence
- Reading and writing numerals to 5, 10, 20, 120, or 1,000
- Comparing quantities to 10, 100, or 1,000; using >, =, and < symbols to record comparisons
- Structuring 5 and 10
- Composing and decomposing numbers from 11 to 19 into 10 ones and some more
- Skip-counting by 10s and 100s within 1,000
- Understanding the place value structure of 2- or 3-digit numbers
- Mentally finding 10 more or 10 less than any 2-digit number without having to count
- Mentally adding or subtracting 10 or 100 to or from a given number 100–900

## Operations: Basic Addition & Subtraction

This volume focuses on the skills and concepts needed to fluently add and subtract within 5, 10, and 20. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one of more of the following:

- Representing addition and subtraction with objects, drawings, verbal explanations, expressions, and equations
- Structuring 5 and 10
- Counting on to add and counting back to subtract
- Developing and using strategies (other than counting on and counting back) to add and subtract within 5, 10, or 20
- Fluently adding and subtracting with sums and minuends to 5, 10, or 20
- Understanding and applying the commutative and associative properties of addition
- Solving subtraction by finding an unknown addend
- Solving for the unknown in addition and subtraction equations

## Base Ten Operations: Addition & Subtraction with Multi‑Digit Numbers

This volume focuses on the place value skills and concepts needed to add and subtract multi-digit numbers with understanding and fluency. More specifically, the collection of resources offered in this volume can be used to provide intervention for students struggling with one of more of the following:

- Reading, writing, comparing, and understanding 2- and 3-digit numbers
- Grouping and counting objects by 10s and 1s, or by 100s, 10s, and 1s
- Counting forward and backward by 10s on and off the decade (200, 210, 220, 230, and so on; 204, 214, 224, 234, and so on)
- Counting forward and backward by 100s on and off the century (100, 200, 300, and so on; 112, 212, 312, and so on)
- Mentally adding or subtracting 10 or 100 from any number in the range of 100–900
- Developing accurate, effective and efficient strategies for adding and subtracting 2- and 3-digit numbers

## Addition & Subtraction Word Problems

This volume focuses on the skills and concepts required for students to unpack and solve addition and subtraction story problems within 10, 20, and 100. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one or more of the following:

- Understanding the operations of addition and subtraction
- Representing addition and subtraction with objects, fingers, mental images, drawings, verbal explanations, expressions, or equations
- Solving addition and subtraction story problems within 10 involving situations of adding to, taking from, putting together, and taking apart
- Solving addition and subtraction story problems with sums and minuends to 20 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions
- Solving one- and two-step addition and subtraction story problems with sums and minuends to 100 involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions

Although story problem contexts may make it easier for some students to understand the operations of addition and subtraction, the resources in this volume are not designed to facilitate the development of fluency with basic addition and subtraction facts or proficiency with double-digit addition and subtraction per se. Bridges Intervention Volumes 2 and 3 are written to address these skills and concepts, and may be used in conjunction with Volume 4 to provide comprehensive intervention for students struggling with one or more aspects of addition and subtraction within 100.

## Operations: Basic Multiplication & Division

This volume focuses on the skills and concepts needed to fluently multiply and divide within 100. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one of more of the following:

- Understanding the operations of multiplication and division, and how they relate to each other
- Fluently multiplying and dividing within 100 using efficient mental strategies
- Recalling from memory all products of two 1-digit numbers
- Understanding and applying the commutative and distributive properties of multiplication
- Solving division problems by finding an unknown factor
- Identifying patterns among basic multiplication facts and patterns in the multiplication table
- Relating area to the operation of multiplication

## Base Ten Operations: Multiplication-Division of Multi-Digit Numbers

This volume focuses on the skills and concepts students need to develop efficient, accurate, and flexible strategies for multiplying and dividing multi-digit numbers. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one of more of the following:

- Understanding and applying the concept of multiplicative comparison
- Understanding and applying the place value concepts associated with multi-digit multiplication and division
- Multiplying a whole number of up to four digits by a 1-digit whole number, or multiplying two 2-digit numbers, using strategies based on place value, the properties of operations, or the relationship between multiplication and division
- Dividing a whole number of up to four digits by a 1- or 2-digit number using strategies based on place value, the properties of operations, or the relationship between multiplication and division
- Using rectangular arrays, area models, T-charts (ratio tables), and equations to model and solve multi-digit multiplication and division problems
- Understanding and applying the commutative, associative, and distributive properties of multiplication
- Using the standard algorithm with fluency to multiply multi-digit whole numbers

## Multiplication & Division Word Problems

This volume focuses on the skills and concepts required for students to unpack and solve multiplication and division story problems, as well as multi‑step story problems requiring the use of all four operations. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are struggling with one or more of the following:

- Solving multiplication and division story problems involving situations of equal groups and arrays
- Solving story problems involving multiplicative comparison using multiplication or division
- Solving two‑step and multi‑step story problems using addition, subtraction, multiplication, or division, including problems in which remainders must be interpreted
- Representing story problems using equations with a letter standing for the unknown quantity
- Solving story problems involving distances, intervals of time, liquid volumes, area, and money

Although story problem contexts may make it easier for some students to understand the operations of multiplication and division, the resources in this volume are not designed to facilitate the development of fluency with basic multiplication and division facts or proficiency with multi‑digit multiplication and division. In fact, much of the work in Volume 7 rests on the assumption that students will possess a reasonable amount of skill with the basic facts and multi‑digit computation. This is especially true for Modules 6–8, in which two‑step and multi‑step problems add extra layers of complexity and challenge. Bridges Intervention Volumes 5 and 6 are written to address basic facts and multi‑digit multiplication and division, and should be used to help Tier 2 students develop computational and procedural fluency with these two operations prior to Volume 7.

## Adding, Subtracting & Making Sense of Fractions

The 13 modules in this volume help students to make sense of fractions and mixed numbers, and to develop strategies for comparing, adding, and subtracting fractions with like and unlike denominators. It can be used to provide intervention for students who are struggling with one of more of the following:

- Representing fractions as equal parts of a whole area
- Representing fractions using bar models and number lines
- Identifying and generating equivalent fractions
- Identifying fractions and whole numbers that are equal
- Identifying fractions and mixed numbers that are equal
- Comparing fractions
- Adding and subtracting fractions
- Rewriting fractions as equivalent fractions with a common denominator to compare, add, and subtract them
- Solving story problems involving addition and subtraction of fractions

Note that this volume does not address multiplication or division with fractions.

## Money & Decimals

This volume focuses on a collection of skills from Grades 2, 4, and 5 related to money and decimals. The resources in this volume can be used to provide intervention for students who are struggling with one of more of the following:

- Recognizing the names and values of coins
- Finding the value of a collection of coins
- Solving problems about money
- Connecting decimal fractions with decimal numbers
- Adding decimal fractions
- Understanding and comparing decimal numbers
- Adding decimal numbers