# 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

This volume focuses 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 working on 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 from 100 to 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 working on 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 Multidigit Numbers

This volume focuses on the place value skills and concepts needed to add and subtract multidigit numbers with understanding and fluency. More specifically, the collection of resources offered in this volume can be used to provide intervention for students working on 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 to 900
- Developing accurate, effective and efficient strategies for adding and subtracting 2- and 3-digit numbers

## Addition & Subtraction Problem Situations

This volume focuses on the skills and concepts required for students to unpack and solve addition and subtraction problem situations 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 working on 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 problem situations within 10 involving situations of adding to, taking from, putting together and taking apart
- Solving addition and subtraction problem situations 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 problem situations 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 problem situation 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 working on 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 working on 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

## Multiplication & Division of Multidigit Numbers

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

- Understanding and applying the concept of multiplicative comparison
- Understanding and applying the place value concepts associated with multidigit 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 multidigit multiplication and division problems
- Understanding and applying the commutative, associative and distributive properties of multiplication
- Using the standard algorithm with fluency to multiply multidigit whole numbers

## Multiplication & Division Problem Situations

This volume focuses on the skills and concepts required for students to unpack and solve multiplication and division problem situations, as well as multistep problem situations 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 working on one or more of the following:

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

Although problem situation 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 multidigit 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 multidigit computation. This is especially true for Modules 6 through 8, in which two‑step and multistep 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 students develop computational and procedural fluency with these two operations prior to Volume 7.

## Adding, Subtracting & Making Sense of Fractions

This volume helps students to make sense of fractions and mixed numbers, and to develop strategies for comparing, adding and subtracting fractions with like and unlike denominators. More specifically, the collection of resources offered in this volume can be used to provide intervention for students who are working on 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 problem situations 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. More specifically, the collection of resources in this volume can be used to provide intervention for students who are working on 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