Bridges in Mathematics,
2nd edition

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.

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.

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.

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 (2-D) 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.

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.

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 design solar homes.