# Bridges By Grade

## Bridges by Grade

Bridges provides the guidance, materials, and tools for building mathematical thinkers.

Select a grade level to learn more.

## Kindergarten

### Overview

Students focus intensively on the two critical areas specified by the Common Core State Standards for Mathematics:

- Representing and comparing whole numbers
- Describing shapes and space

Six of eight units are devoted to number and operations. They help students learn to use numbers, including written numerals, to represent quantities and solve problems; count out a given number of objects; compare sets or numerals; and model simple joining and separating situations with objects, fingers, words, actions, drawings, numbers, and equations.

The remaining two units focus on geometry. They invite students to describe and analyze the attributes of shapes in the world around them; find, count, draw, build, and compare shapes; and fit shapes together to make other shapes and complete puzzles.

### Content Outline

#### Unit 1 Numbers to Five & Ten

- Sorting Shoes
- Friendly Fives
- Friendly Tens
- Using Structures & Patterns

#### Unit 2 Numbers to Ten

- Dots to Ten
- Introducing the Number Rack
- Five & Some More
- Composing & Decomposing Shapes

#### Unit 3 Bikes & Bugs: Double, Add & Subtract

- Bicycle Doubles
- Adding & Subtracting Ones
- Add, Subtract & Double It!
- Put Them in Order

#### Unit 4 Paths to Adding, Subtracting & Measuring

- Paths: The Number Line
- Counting, Adding & Subtracting with Forest Animals
- Comparing & Measuring Length
- Fives & Ones with Money

#### Unit 5 Two-Dimensional Geometry

- Exploring Shapes
- Circles, Squares, Triangles & Rectangles
- Constructing & Drawing Shapes
- Sorting, Comparing, Composing & Decomposing Shapes

#### Unit 6 Three-Dimensional Shapes & Numbers Beyond Ten

- What Do You Know About Three-Dimensional Shapes?
- More Three-Dimensional Shapes
- Exploring the Teen Numbers
- Combinations to Ten

#### Unit 7 Weight & Place Value

- How Heavy? Weight & Number
- Tens & Ones to Twenty
- Addition & Subtraction Story Problems
- Counting by Tens & Ones

#### Unit 8 Computing & Measuring with Frogs & Bugs

- Catching, Counting & Comparing
- Frogs: Estimating & Measuring
- Tens & Ones
- Addition & Subtraction Equations

#### Number Corner

- Calendar Grid
- Calendar Collector
- Days in School
- Computational Fluency
- Number Line

## Grade 1

### Overview

Students focus intensively on the four critical areas specified by the Common Core State Standards for Mathematics in Grade One:

- Addition and subtraction within 20
- Whole number relationships and place value
- Linear measurement in non-standard units
- Reasoning with shapes and their attributes

Four of eight units are devoted to addition and subtraction within 20. They help students gain fluency with facts to 10 and develop increasingly sophisticated strategies for solving addition and subtraction combinations to 20. During these units, students model, solve, and pose a wide variety of word problems to construct meaning for the operations of addition and subtraction, as well as an understanding of how the two are connected.

Two of the eight units focus on place value, extending the counting sequence to 120 as students learn to think of whole numbers between 10 and 100 in terms of tens and ones. Much of the work in these units engages first graders in developing, discussing, and using efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10.

The remaining two units revolve around geometry and measurement. The geometry unit challenges children to identify, describe, construct, draw, compare, compose, and sort shapes. Students also learn about fractions in the context of two-dimensional shapes. The measurement unit provides ample opportunity for students to develop an understanding of the meaning and processes of measuring time and length as they conduct duration experiments; construct, fly, and measure the flight distances of paper gliders; and investigate some of the many ways in which they have grown and changed since they were born.

### Content Outline

#### Unit 1 Numbers All Around Us

- Counting & Data with Popsicles
- Meet the Number Rack
- Part-Part-Whole to Ten
- Adding & Subtracting to Ten with the Number Rack

#### Unit 2 Developing Strategies with Dice & Dominoes

- Counting, Comparing & Adding with Dominoes
- Fact Families & Story Problems
- Introducing Fact Strategies
- Counting by Fives & Tens

#### Unit 3 Adding, Subtracting, Counting & Comparing

- Single-Digit Sums
- Combinations with the Number Rack
- Tens & Teens
- Exploring Equations

#### Unit 4 Leapfrogs on the Number Line

- Adding & Subtracting on the Life-Sized Number Line
- Jumping by Fives & Tens
- Jumping by Fives & Tens on the Open Number Line
- Measuring, Comparing & Subtracting with Penguins

#### Unit 5 Geometry

- Introducing Two-Dimensional Shapes
- Introducing Three-Dimensional Shapes
- Putting Shapes Together & Taking Them Apart
- Sorting & Graphing Shapes

#### Unit 6 Figure the Facts with Penguins

- Story Problems for Basic Addition & Subtraction
- Combinations & Story Problems
- Solving for the Unknown in Penguin Stories
- Measuring & Comparing Emperor & Little Blue Penguins

#### Unit 7 One Hundred & Beyond

- Grouping Sticks & Bundles Beyond One Hundred
- Hansel & Gretel’s Path on the Number Line
- Adding & Subtracting Two-Digit Numbers with Hansel & Gretel
- Place Value with Money

#### Unit 8 Changes, Changes

- Time & Duration
- Patterns, Structure & Change
- Measurement & Data with Paper Gliders
- Measuring our Growth

#### Number Corner

- Calendar Grid
- Calendar Collector
- Days in School
- Computational Fluency
- Number Line

## Grade 2

### Overview

Students focus intensively on the four critical areas specified by the Common Core State Standards for Mathematics in Grade Two:

- Extending understanding of base-ten notation
- Building fluency with addition and subtraction
- Using standard units of linear measurement
- Describing and analyzing shapes

The first unit revisits and extends addition and subtraction within 20, helping to ensure that second graders operate with understanding and fact fluency from the start of the school year.

Units 2, 3, 5, and parts of Unit 7, are devoted to place value and multi-digit addition and subtraction. During these units, students learn to count by fives, tens, and multiples of hundreds, tens and ones; read, write, and compare numbers to 1,000; and develop fluency with addition and subtraction to 100 as they solve and pose a wide variety of word problems. Later in the year, the children use concrete models and sketches, as well as strategies based on place value, properties of operations, and the relationship between addition and subtraction, to add and subtract to 1,000.

Unit 6 revolves around geometry, building foundations for understanding area, volume, congruence, similarity, and symmetry as students investigate, describe, build, draw, combine, decompose, and analyze two- and three-dimensional shapes.

Unit 4, and the first part of Unit 7, focus on linear measurement, as students construct their own rulers; estimate and measure in inches, feet, yards, centimeters, and meters; and solve problems that involve adding, subtracting, and comparing lengths.

Unit 8 revisits linear measurement in the context of science and engineering as students make and test cardboard ramps of different kinds to investigate some of the factors that cause marbles to roll farther and faster. In the process, they generate data by measuring marble roll distances multiple times, pool their data, and enter it on line plots to better see, understand, and analyze how manipulating the different variables affects the outcomes.

### Content Outline

#### Unit 1 Figure the Facts

- Sorting & Graphing
- Number Facts with the Number Rack
- Introducing Addition & Subtraction Strategies
- Fluency with Addition Facts to Twenty

#### Unit 2 Place Value & Measurement with Jack's Beanstalks

- Counting & Modeling Two- & Three-Digit Numbers
- Measuring Jack’s Giant Beans with Tens
- Adding on the Open Number Line
- Thinking in Twos

#### Unit 3 Addition & Subtraction Within One Hundred

- Tens & Ones
- Adding & Subtracting on the Number Line
- Present & Parcel Story Problems with Two-Digit Numbers
- Data & the Many Colors Project

#### Unit 4 Measurement

- Inches & Feet
- Inches, Feet & Yards
- Proportion & Fractions with a Giant
- Thinking in Threes

#### Unit 5 Place Value to One Thousuand

- Counting to One Thousand
- Place Value with Money
- Multiples of Ten, One Hundred & One Thousand
- Sequences & Patterns

#### Unit 6 Geometry

- Attributes of Two-Dimensional Shapes
- Exploring Area & Arrays
- Composing & Decomposing Patchwork Shapes
- Patchwork Fractions

#### Unit 7 Measurement, Fractions & Multi-Digit Computation with Hungry Ants

- Army Ants: Length in Metric Units
- Ant Treats: Division & Fractions
- Adding & Subtracting Three-Digit Numbers
- Writing & Solving Story Problems

#### Unit 8 Measurement, Data & Multi-Digit Computation with Marble Rolls

- Revisiting Place Value & Three-Digit Computation
- Building Marble Rolls & Collecting Data
- Collecting & Analyzing More Marble Roll Data
- Student-Conducted Surveys

#### Number Corner

- Calendar Grid
- Calendar Collector
- Daily Rectangle
- Computational Fluency
- Number Line

## Grade 3

### Overview

Students focus intensively on the four critical areas specified by the Common Core State Standards for Mathematics in Grade Three:

- Developing understanding of multiplication and division and strategies for multiplication and division within 100
- Developing understanding of fractions, especially unit fractions (fractions with numerator 1)
- Developing understanding of the structure of rectangular arrays and of area
- Describing and analyzing two-dimensional shapes

The first unit reviews and extends work with addition and subtraction as students review facts, look for patterns, and work with larger numbers. Unit 2 transitions to multiplication by having students use a variety of rich contexts (arrays of stamps, groups of windows, and a coral reef) to develop and refine multiplication strategies and models. Unit 3 returns to addition and subtraction, this time focusing on strategies for computing with larger numbers.

In Units 4 and 5, students explore measurement, fractions, division, and multiplication of larger numbers. They estimate and make measurements in different units; explore unit fractions and equivalent fractions, and begin adding and subtracting fractions; they connect multiplication to division and extend multiplication strategies to larger numbers. Their work with multiplication develops a strong understanding of area.

Unit 6 focuses on geometry, as students investigate, draw, and build two-dimensional shapes, using their properties to classify and analyze these shapes. They also connect geometry to fractions as they express the area of a shape as a unit fraction of the whole. Unit 7 brings together and extends many of the skills and concepts addressed in earlier units as students solve challenging problems that involve calculating with multi-digit numbers. They explore algorithms for addition and subtraction and dig deeper into division. Students develop strategies and models for division, many of which are based on their work with multiplication.

Unit 8 integrates mathematics and science, with a primary focus on designing and building model bridges. Students test the strength of their model bridges in systematic ways to collect data. Then they graph and analyze the data, finding the range and mean, to make conjectures and draw conclusions about effective bridge design and construction.

### Content Outline

#### Unit 1 Addition & Subtraction Patterns

- Community Building & Addition Facts to Twenty
- Subtraction Facts to Twenty
- Double-Digit Addition
- Story Problems & Strategies

#### Unit 2 Introduction to Multiplication

- Multiplication in Context
- Multiplying with Arrays & Number Lines
- Ratio Tables & the Multiplication Table
- Story Problems with Graphs & Multiple Operations

#### Unit 3 Multi-Digit Addition & Subtraction

- Rounding &

Multi-Digit Addition - Multi-Digit Subtraction
- Estimating to Add

& Subtract - Exploring the Algorithms

for Addition

& Subtraction

#### Unit 4 Measurement & Fractions

- Measuring Time & Mass
- Measuring Volume & Solving Measurement Story Problems
- Fractions as Fair Shares
- Fractions on a Line Plot

#### Unit 5 Multiplication, Division & Area

- Linking Multiplication & Division
- Multiplication & Division Families
- Division Practice
- Introducing Area

#### Unit 6 Geometry

- Investigating Polygons
- Quadrilaterals
- Perimeter & Area
- Shapes & Fractions

#### Unit 7 Extending Multiplication & Fractions

- Multiplication Beyond the Basics
- One- by Two-Digit Multiplication
- Fractions as Parts of a Whole & Parts of a Set
- Fractions at Work

#### Unit 8 Bridge Design & Construction: Data Collection and Analysis

- Introducing Bridges
- Investigating Structures in Bridges
- Planning, Building & Analyzing Bridges
- Demonstrating Our Learning About Bridges

#### Number Corner

- Calendar Grid
- Calendar Collector
- Computational Fluency
- Number Line
- Solving Problems

## Grade 4

### Overview

Students focus intensively on the three critical areas specified by the Common Core State Standards for Mathematics in Grade Four:

- Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends
- Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers
- Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry

The first two units focus on multiplication and multiplicative thinking. Unit 1 reviews and extends multiplication work from third grade and examines factors and products, as well as prime and composite numbers. Unit 2 delves deeper as students explore and extend strategies, concepts, and models related to multi-digit multiplication. Unit 3 utilizes a variety of tools to model, read, write, compare, order, compose and decompose fractions and decimals.

Units 4, 6 and 7 focus on fractions, decimals, division, and more multiplication. Various models help students understand more about operations with fractions and fraction equivalence, as well as the relationship between fractions and decimals. Students also discover the relationships between multiplication and division as they see that many multiplication strategies also apply to division problems. They solve division problems with and without remainders and begin exploring multiplication and division of simple fractions.

Unit 5 focuses on geometry and extends students’ understandings of area, volume, and symmetry. Students investigate, draw, and build two-dimensional shapes and the properties of those shapes to classify and analyze them. They also learn to use protractors to measure and construct angles.

Unit 8 integrates many key skills and concepts in the context of science and engineering by giving students the opportunity to design playgrounds. In the process, they generate and analyze data, and use a line plot to represent that data.

### Content Outline

#### Unit 1 Multiplicative Thinking

- Models for Multiplication & Division
- Primes & Composites
- Multiplicative Comparisons & Equations
- Measurement Experiences

#### Unit 2 Multi-Digit Multiplication

& Early Division

- Building Multiplication Arrays
- Arrays & Ratio Tables
- Multiplication Stories & Strategies
- Early Division with Remainders

#### Unit 3 Fractions & Decimals

- Equivalent Fractions
- Comparing, Composing & Decomposing Fractions & Mixed Numbers
- Introducing Decimals
- Fractions & Decimals

#### Unit 4 Addition, Subtraction & Measurement

- Place Value & the Traditional Addition Algorithm
- The Traditional Subtraction Algorithm
- Measurement
- Measurement & Data Displays

#### Unit 5 Geometry & Measurement

- Measuring Angles
- Polygons & Symmetry
- Area & Perimeter
- Angles in Motion

#### Unit 6 Multiplication & Division, Data & Fractions

- Multiplication & Division Strategies
- Revisiting Area & Perimeter
- Line Plots, Fractions & Division
- More Division

#### Unit 7 Reviewing & Extending Fractions, Decimals & Multi-Digit Multiplication

- Comparing Fractions & Writing Equivalent Fractions
- Decimals & Decimal Fractions
- Introducing the Standard Multiplication Algorithm
- Extending the Standard Multiplication Algorithm

#### Unit 8 Playground Design

- Introducing Playground Design
- Making Decisions
- Using Scale Models for Our Playground & Field
- Building Model Playgrounds

#### Number Corner

- Calendar Grid
- Calendar Collector
- Computational Fluency
- Problem Strings
- Solving Problems

## Grade 5

### Overview

Students focus intensively on the three critical areas specified by the Common Core State Standards for Mathematics in Grade Five:

- Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions)
- Extending division to 2-digit divisors, integrating decimal fractions into the place value system, developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations
- Developing understanding of volume

The first unit is focused on volume, and includes a review of multiplication facts and multi-digit multiplication strategies. In Unit 2, students use what they know about equivalent fractions to add and subtract fractions. Unit 3 extends students’ understandings of place value and the properties of operations to help students develop powerful strategies for computing fluently with decimals. In Unit 4 they refine powerful multiplication and division strategies, including the array model and the standard algorithm for multiplication.

In Unit 5 students learn to multiply and divide fractions. Unit 6 introduces new geometric concepts, including coordinate graphing and the use of hierarchies to classify two-dimensional shapes by their properties.

In Unit 7 students develop accurate and efficient strategies for dividing whole numbers, decimals, and fractions (unit fractions by whole numbers, and whole numbers by unit fractions). Unit 8 integrates science, engineering, and math. In this final unit, students apply the understandings and skills they have developed over the year as they study solar energy and designing solar homes.

### Content Outline

#### Unit 1 Expressions, Equations & Volume

- Multiplication & Volume
- Factors, Multiples & the Associative Property
- Multiplication Strategies
- From Multiplication to Division

#### Unit 2 Adding & Subtracting Fractions

- Adding & Subtracting Fractions
- Fractions as Operators & Quotients
- Common Denominators
- LCMs & GCFs

#### Unit 3 Place Value & Decimals

- Whole Number & Decimal Place Value
- Adding & Subtracting Decimals
- Conversions
- Division & The Area Model

#### Unit 4 Multiplying & Dividing Whole Numbers & Decimals

- Multiplication & Division Strategies
- More Multiplication & Division Strategies
- From Array to Algorithm
- Multiplying to Divide

#### Unit 5 Multiplying & Dividing Fractions

- Multiplying Whole Numbers by Fractions
- Multiplying Fractions by Fractions
- More Fraction by Fraction Multiplication
- Dividing Fractions & Whole Numbers

#### Unit 6 Graphing, Geometry & Volume

- Graphing Ordered Pairs
- Classifying Polygons
- Volume
- Banners & Flags

#### Unit 7 Division & Decimals

- Division of Fractions & Whole Numbers
- Division Interpretations & Strategies
- Powers of Ten
- Decimal Multiplication & Division

#### Unit 8 Solar Design

- Investigating Solar Energy
- Investigating Passive Solar Design
- Designing Solar Homes
- Finishing Our Models

#### Number Corner

- Calendar Grid
- Calendar Collector
- Computational Fluency
- Problem Strings
- Solving Problems