## Scope and Sequence

### (with Spanish)

### Narrative

The big ideas in grade 2 include: extending understanding of the base-ten number system, building fluency with addition and subtraction, using standard units of measure, and describing and analyzing shapes.

The mathematical work for grade 2 is partitioned into 9 units:

- Adding, Subtracting, and Working with Data
- Adding and Subtracting within 100
- Measuring Length
- Addition and Subtraction on the Number Line
- Numbers to 1,000
- Geometry, Time, and Money
- Adding and Subtracting within 1,000
- Equal Groups
- Putting it All Together

In these materials, particularly in units that focus on addition and subtraction, teachers will find terms that refer to problem types, such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the Mathematics Glossary section of the Common Core State Standards.

### Unit 1: Sumemos, restemos y trabajemos con datos

**Unit Learning Goals**

- Students represent and solve story problems within 20 through the context of picture and bar graphs that represent categorical data. Students build toward fluency with addition and subtraction.

In this unit, students begin the year-long work to develop fluency with sums and differences within 20, building on concepts of addition and subtraction from grade 1. They learn new ways to represent and solve problems involving addition, subtraction, and categorical data.

In grade 1, students added and subtracted within 20 using strategies based on properties of addition and place value. They developed fluency with sums and differences within 10. Students also gained experience in collecting, organizing, and representing categorical data.

Here, students are introduced to picture graphs and bar graphs as a way to represent categorical data. They ask and answer questions about situations described by the data. The structure of the bar graphs paves the way for a new representation, the tape diagram.

Students learn that tape diagrams can be used to represent and make sense of problems involving the comparison of two quantities. The diagrams also help to deepen students’ understanding of the relationship between addition and subtraction.

This opening unit also offers opportunities to introduce mathematical routines and structures for centers, and to develop a shared understanding of what it means to do math and to be a part of a mathematical community.

#### Section A: Sumemos y restemos hasta 20

**Standards Alignments**

Addressing | 2.NBT.B.5, 2.OA.B.2 |

**Section Learning Goals**

- Build toward fluency with adding within 100.
- Build toward fluency with subtracting within 20.

This opening section gives teachers opportunities to assess students’ fluency with addition and subtraction facts within 10 and how they approach adding and subtracting.

The first several lessons focus on making a ten as a strategy to add and subtract, which helps students gain fluency with facts within 20 and supports the work with larger numbers (such as composing and decomposing numbers as a way to add and subtract). In the last lesson of the section, students use strategies learned in grade 1 to add within 50.

\(\hspace{3cm}\)

\(10- 5 = \underline{\hspace{1 cm}}\)

\(5 + \underline{\hspace{1 cm}}=10\)

\(2 + \underline{\hspace{1 cm}}=10\)

\(10 - 8 = \underline{\hspace{1 cm}}\)

Some activities take place in centers, enabling teachers to also introduce routines and structures while helping students develop mental strategies for adding and subtracting.

#### Section B: Formas de representar datos

**Standards Alignments**

Addressing | 2.MD.D.10, 2.NBT.B.5, 2.OA.B.2 |

**Section Learning Goals**

- Interpret picture and bar graphs.
- Represent data using picture and bar graphs.
- Solve one- and two-step problems using addition and subtraction within 20.

In this section, students explore situations and problems that involve categorical data and learn new ways to represent such data.

Students begin by representing data about their class in a way that makes sense to them. Then, they are introduced to picture graphs and bar graphs. Students learn the conventions of these graphs as they create them. They discuss the types of questions that can be asked and answered by the graphs, including those that require combining and comparing different categories.

#### Section C: Diagramas para comparar

**Standards Alignments**

Addressing | 2.MD.D.10, 2.NBT.A.2, 2.NBT.B.5, 2.OA.A.1, 2.OA.B.2 |

**Section Learning Goals**

- Make sense of and interpret tape diagrams.
- Represent and solve Compare problems with unknowns in all positions within 100.

Students have previously represented and reasoned about quantities in story problems. In grade 1, students compared quantities using diagrams with discrete partitions. In the previous section, they reasoned about quantities in bar graphs. Here, students learn to use tape diagrams as another way to make sense of the relationship between two quantities and between addition and subtraction.

Students explore Compare story problems with an unknown difference, an unknown larger number, or an unknown smaller number. Tape diagrams help students to visualize these structures and support them in reasoning about strategies to use to solve problems, such as counting on or counting back. The table highlights the different types of problems in this section.

difference unknown | bigger unknown | smaller unknown |
---|---|---|

Lin counted 28 boats. Diego counted 32 boats. How many more boats did Diego count? | Lin found 28 more shells than Diego. Diego found 32 shells. How many shells did Lin find? | Lin saw 32 starfish. Diego saw 28 fewer starfish than Lin. How many starfish did Diego see? |

Students also write equations to reason about questions that ask “how many more?” and “how many less?” They recognize that different equations and diagrams can be used to represent the same difference between two numbers.

Estimated Days: 14 - 18

### Unit 2: Sumemos y restemos hasta 100

**Unit Learning Goals**

- Students add and subtract within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction. They then use what they know to solve story problems.

Previously, students added and subtracted numbers within 100 using strategies they learned in grade 1, such as counting on and counting back, and with the support of tools such as connecting cubes. In this unit, they add and subtract within 100 using strategies based on place value, the properties of operations, and the relationship between addition and subtraction.

Students begin by using any strategy to find the value of sums and differences that do not involve composing or decomposing a ten. They are then introduced to base-ten blocks as a tool to represent addition and subtraction and move towards strategies that involve composing and decomposing tens.

Students develop their understanding of grouping by place value, and begin to subtract one- and two-digit numbers from two-digit numbers by decomposing a ten as needed. They apply properties of operations and practice reasoning flexibly as they arrange numbers to facilitate addition or subtraction.

For example, students compare Mai and Lin’s methods for finding the value of \(63-18\).

Mai’s Way

\(63 - 18\)

Lin’s Way

\(63 - 18\)

At the end of the unit, students apply their knowledge of addition and subtraction within 100 to solve one- and two-step story problems of all types, with unknowns in all positions. To support them in reasoning about place value when adding and subtracting, students may choose to use connecting cubes, base-ten blocks, tape diagrams, and other representations learned in earlier units and grades.

#### Section A: Sumemos y restemos

**Standards Alignments**

Addressing | 2.MD.D.10, 2.NBT.A.2, 2.NBT.B.5, 2.NBT.B.9, 2.OA.A.1, 2.OA.B.2 |

**Section Learning Goals**

- Add and subtract within 100 using strategies based on place value and the relationship between addition and subtraction. Problems in this section are limited to the problems like 65 – 23, where decomposing a ten is not required.

In this section, students find the value of unknown addends using methods that are based on place value and are introduced to base-ten blocks. They continue to rely on the relationship between addition and subtraction to solve problems involving differences.

Students begin by solving Compare story problems. They use any methods and tools that make sense to them—including diagrams and connecting cubes—to find differences of two-digit numbers.

*Lin and Clare used cubes to make trains.
What do you notice? What do you wonder?*

Students then analyze the structure of base-ten blocks and use them to find unknown addends (MP7). Unlike connecting cubes, base-ten blocks cannot be pulled apart, which helps emphasize the structure of two-digit numbers in base ten.

To reason about an unknown addend, they may add tens and ones to the known addend until they reach the value of the sum. They may also start with the total amount and subtract tens from tens and ones from ones to reach the known addend. The numbers encountered here do not require students to decompose a ten when they subtract by place value.

#### Section B: Descompongamos para restar

**Standards Alignments**

Addressing | 2.NBT.B.5, 2.NBT.B.6, 2.NBT.B.9, 2.OA.B.2 |

**Section Learning Goals**

- Subtract within 100 using strategies based on place value, including decomposing a ten, and the properties of operations.

In this section, students subtract one- and two-digit numbers from two-digit numbers within 100. To reason about differences of two numbers, they use methods based on place value, base-ten blocks and diagrams, and properties of operations. The numbers here require students to decompose a ten when subtracting by place.

Students also make sense of different representations of subtraction by place, including those that show their peers’ reasoning. For example, to find the value of \(63-18\), students might use base-ten blocks or drawings to represent tens and ones. In this case, they might decompose 1 ten from 63 and exchange it for 10 ones, making 5 tens and 13 ones. From here, some students may first take away 8 ones, and then 1 ten. Others may take away 1 ten, then 8 ones.

When students discuss different approaches and explain why they result in the same value, they deepen their understanding of the properties of operations and place value.

\(63 - 18\)

The reasoning here builds a foundation for students to understand the standard algorithm for subtraction, but students should not be encouraged to use the notation for standard algorithm at this point. Allow them to build conceptual understanding by reasoning with base-ten blocks and drawings and articulating their thinking.

#### Section C: Representemos y resolvamos problemas-historia

**Standards Alignments**

Addressing | 2.NBT.B.5, 2.NBT.B.6, 2.OA.A.1, 2.OA.B.2 |

**Section Learning Goals**

- Represent and solve one- and two-step problems involving addition and subtraction within 100, including different problem types with unknowns in all positions.

This section allows students to apply their knowledge to solve story problems that involve addition and subtraction within 100. The story problems include all types—Add To, Take From, Put Together/Take Apart, and Compare— and have unknowns in all positions.

Previously, students worked with diagrams that represent Compare problems. Throughout this section, students also make sense of diagrams that could represent Put Together/Take Apart story problems.

*Clare and Han are playing a game with seeds.
Clare has 54 seeds on her side of the board.
Han has 16 seeds on his side.
How many seeds are on the board in all?
Which diagram matches this story? Explain your match to your partner.*

As students relate quantities in context and diagrams that represent them, they practice reasoning quantitatively and abstractly (MP2).

Throughout the section, students are invited to interpret and solve problems in the ways that make sense to them (MP1). Math tools such as connecting cubes and base-ten blocks should be made available to encourage methods based on place value and the properties of operations to solve the problems.

Estimated Days: 12 - 16