## Scope and Sequence

### Narrative

The big ideas in kindergarten include: representing and comparing whole numbers, initially with sets of objects; understanding and applying addition and subtraction; and describing shapes and space. More time in kindergarten is devoted to numbers than to other topics.

The mathematical work for kindergarten is partitioned into 8 units:

- Math in Our World
- Numbers 1–10
- Flat Shapes All Around Us
- Understanding Addition and Subtraction
- Composing and Decomposing Numbers to 10
- Numbers 0–20
- Solid Shapes All Around Us
- 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: Math in Our World

**Unit Learning Goals**

- Students recognize numbers and quantities in their world.

In this unit, students explore mathematical tools and notice numbers and quantities around them, while teachers gather information about students’ counting skills and understanding of number concepts.

Students enter kindergarten with a range of counting experiences, concepts, and skills. This unit is designed to be accessible to all learners regardless of their prior experience. To that end, no counting is required for students to engage in the activities in the first three sections, though students may choose to count. Students also have opportunities to work with math tools and topics related to geometry, measurement, and data through a variety of centers.

In the last section, students count collections of objects and groups of people, answering “how many of _____ are there?” questions. These questions reinforce the idea that counting is a way to tell how many objects there are. Students are expected to count up to 10 objects by the time they begin the next unit, which will focus more deeply on numbers 1–10.

The unit is also designed to give students time to learn the structures and routines for centers, to create norms for classroom learning, and to begin to build a mathematical community. The content and timing of the lessons at the beginning of the unit are calibrated to make this possible.

To gather information about students’ counting and number concepts, consider asking individual students to count a small group of objects and observing the skills or understandings listed in the provided checklist. The end-of-unit assessment, a one-on-one interview, is another opportunity to find out what students know and can do. This assessment is not necessary for those who have demonstrated the skills on the checklist throughout the unit.

#### Section A: Explore Our Math Tools

**Standards Alignments**

Addressing | K.CC, K.G, K.G.B |

**Section Learning Goals**

- Explore and use math tools.
- Share mathematical ideas with a partner.

In this section, students build a shared understanding of what it means to do math and to be a part of a mathematical community, where everyone’s contributions are valued. They collaborate to create norms for their work together. They are also encouraged to share their ideas and listen to others’, make connections between their work and their home life, and to see themselves as productive mathematical thinkers.

Students also interact with the tools that they will use in math activities and centers throughout the year. They have the opportunity to freely explore the tools and think of their mathematical purposes before choosing a tool for use in structured activities later in the section and in centers.

Consider taking the time in this section to formatively assess students’ counting concepts and skills, observing students or asking them to count small groups of objects while they work, and using the Sections A-D Checkpoint document from the teacher resource pack.

PLC: Lesson 2, Warm-up, Notice and Wonder: Pattern Blocks

#### Section B: Recognize Quantities

**Standards Alignments**

Addressing | K.CC, K.CC.B.4 |

**Section Learning Goals**

- Recognize and name groups of up to 4 objects and images without counting.

In this section, students continue to explore numbers and quantities in their classroom, focusing on small groups of objects or images they can quantify without counting. They match groups that have the same number of images and notice that the same quantity can be arranged in many different ways. Students continue to develop the language to express these ideas and to listen to ideas of their peers.

Students are sometimes asked to show quantities up to 5 on their fingers. This is a chance to formatively observe if students are comfortable showing quantities on their fingers (any way is acceptable). For example, they may put up 4 fingers to show how many objects there are before saying the number word “four.”

This section provides continued opportunity to formatively assess students’ counting concepts and skills.

PLC: Lesson 9, Activity 2, Introduce Picture Books, Create

#### Section C: Are There Enough?

**Standards Alignments**

Addressing | K.CC |

**Section Learning Goals**

- Answer "are there enough" questions.

In this section, students work on the concept of one-to-one correspondence. They match one object to one person or image to answer “are there enough” questions and to get enough objects. This matching skill will be useful in the next section and in future counting when students match one number word to one object.

“Are there enough” and “can you get enough” questions encourage students to mathematize situations. Look for ways to incorporate these prompts into other parts of the school day, for example, when classroom supplies are being distributed.

PLC: Lesson 10, Activity 2, Are There Enough?

#### Section D: Counting Collections

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.B, K.CC.B.4, K.CC.B.4.a, K.G.B |

**Section Learning Goals**

- Count groups of up to 10 objects.

In this section, students focus on counting up to 10 objects and answering “how many of _____ are there” questions.

They learn a new routine, Questions About Us, and consider the question “how many of us are here today?” The routine offers opportunities to highlight one-to-one matching and the idea of keeping track of what is being counted.

Students also count collections of objects from the classroom or from home. To initiate counting, ask “how many of _____ are there?” instead of saying “count the objects.” This helps to reinforce counting as a way to quantify a collection and the idea of cardinality—that the last number called tells us how many there are.

Students may use counting mats, 5-frames, or other tools to help them count. Representing the numbers 6–10 on a 5-frame, for instance, helps students see the \(5+n\) structure of these numbers. (The 10-frame will be introduced in a future unit.)

Some students may be able to subitize, or recognize how many objects there are without counting. Those who can do so accurately should not be required to count individual objects. Consider differentiating the size of collections students count based on observations of students’ counting.

Included in each lesson is an optional activity to support students in certain aspects of counting—verbalizing the count sequence, one-to-one tagging, and organizing objects to count.

PLC: Lesson 12, Activity 1, Counting Collections

Estimated Days: 16 - 17

### Unit 2: Numbers 1–10

**Unit Learning Goals**

- Students answer “how many” questions, count out, and compare groups within 10. Students write a number to represent how many.

In this unit, students continue to develop counting concepts and skills, including comparing, while learning to write numbers.

Previously, students answered “how many” and “are there enough” questions and counted groups of up to 10 objects. They also learned the structures and routines for activities and centers.

Here, students rely on familiar activity structures to build their counting skills and concepts. First, they count and compare the number of objects, and then do the same with groups of images. The images are given in different arrangements—in lines, arrays, number cube patterns, on 5-frames—to help students connect different representations to the same number.

Use of fingers and 5-frames to represent numbers are emphasized and encouraged because they can help students see the structure of numbers 6–10 as \(5+n\). (Ten-frames will be introduced in a later unit.) Fingers are also helpful for counting and are always available.

In these materials, quantities represented with fingers are shown, from students’ perspective, to start with the left pinky. Numbers 6–10 continue with the thumb on the right hand. When demonstrating numbers on fingers for students, begin with the right pinky so that students see the fingers being held up from left to right.

Students can represent numbers with their fingers in any way, as long as they show the correct number of fingers. It may be helpful to students to hold their fingers down on the table or on their lap to represent 8 and 9.

To compare the number of objects or images, students start by using terms such as “fewer” and “more.” Later, when comparing written numbers, the term “less” is introduced. In general, “less” is used to compare numerals, and “fewer” is used to compare groups of objects. Students may use these terms interchangeably at first, but will develop proficiency with the distinction over time.

#### Section A: Count and Compare Groups of Objects

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.A.3, K.CC.B.4, K.CC.B.4.b, K.CC.B.5, K.CC.C.6 |

**Section Learning Goals**

- Connect quantities with spoken number words.
- Count and compare up to 10 objects and know the number remains the same regardless of the arrangement of the objects.

In this section, students count to answer “how many” questions and develop their understanding of the connection between quantities and spoken number words.

Students are encouraged to use their fingers to count. They may also continue to use any tools and resources from earlier work, such as counting mats and 5-frames, as well as bring objects from home to count. As students count and rearrange objects, students notice that the arrangement of objects does not affect the number of objects (conservation of number). They will continue to build this understanding over time.

Students also develop their comparison skills. They start with quantities that are very different and can be compared visually, such as 7 and 2, and relate the comparisons to the terms “more” and “fewer,” which may be new. (Students do not need to produce grammatically accurate language, but the teacher should use “fewer” or “less” as appropriate in context.)

Display and write the number associated with a quantity whenever possible. Students will begin recognizing, representing, and writing numbers in the second half of the unit.

PLC: Lesson 4, Activity 2, Introduce Shake and Spill: Which Is More?

#### Section B: Count and Compare Groups of Images

**Standards Alignments**

Addressing | K.CC.B, K.CC.B.4, K.CC.B.4.b, K.CC.B.5, K.CC.C.6 |

**Section Learning Goals**

- Connect quantities with spoken number words.
- Count and compare up to 10 images in organized arrangements and know the number remains the same regardless of the order in which the images are counted.

Students begin this section by counting images for the first time. This can be more challenging, as images cannot be rearranged, and students may not have limited experience with keeping track of counted items.

Students encounter groups of images in lines, arrays, 5-frames, number cube arrangements, and on fingers. They may be able to determine the cardinality of some groups of images without counting (subitize), which is a valid way to answer “how many” questions.

Images arranged on 5-frames and images of fingers allow students to work with the structure of “5 and some more.” Repeated experience with this structure can help students see that they can count on from 5 to determine how many images there are.

Here, students also answer “are there enough” questions.

*“Are there enough cartons of milk for each student? How do you know?”*

PLC: Lesson 9, Activity 2, More and Fewer with 5-frames and Fingers

#### Section C: Connect Quantities and Numbers

**Standards Alignments**

Addressing | K.CC, K.CC.A.3, K.CC.B.4, K.CC.B.5, K.CC.C.6 |

**Section Learning Goals**

- Connect quantities with spoken number words and written numbers.
- Understand the relationship between number and quantity.

Previously, students counted and made connections between quantities and spoken number words. In this section, students write numbers to represent quantities. To develop students’ familiarity with written numbers, consider providing a reference sheet with numbers and quantities in 5-frames.

Students also explore new counting tasks: counting images arranged in a circle, and counting objects or drawing images to represent given numbers. Images arranged in a circle are harder to quantify than those in lines, arrays, or frames because there is no defined starting or stopping point. It requires students to develop a method to keep track of which images they have counted.

Creating or drawing a collection with a specified number of items is also more demanding as students need to keep track of the number they are representing and how many they have already counted. In many activities, students have opportunities to look for and make use of structure to help them with the tasks at hand (MP7).

*“Draw a line from each number to the group of dots that it matches.”*

PLC: Lesson 14, Activity 1, Toppings on Pizza

#### Section D: Compare Numbers

**Standards Alignments**

Addressing | K.CC.A.3, K.CC.B.4, K.CC.B.4.c, K.CC.B.5, K.CC.C.6, K.CC.C.7 |

**Section Learning Goals**

- Compare written numbers 1–10.

In this section, students develop their capacity to compare written numbers. As they count, students can see that the numbers get larger and that there is 1 more each time. Here, they determine “1 more” and “1 less” than a given number or group of objects, strengthening their understanding of the relationships between numbers and the foundation for comparing numbers.

Students may compare written numbers in several ways:

- Create representations of each number and use the representations to compare.
- Use number sense (for instance, that 10 is a “big” number) or mental images of numbers (for instance, 4 relates to 4 fingers).
- Use the knowledge of the count sequence: that numbers that come later in the count sequence are greater.

Students who use number sense or mental images may be able to easily compare some numbers but not others. For instance, they may know that 9 is close to 10 or all the fingers in two hands and 4 is associated with fingers in one hand, so 9 is more than 4.

PLC: Lesson 18, Warm-up, Act It Out: Forks for Dinner

Estimated Days: 21 - 22

### Unit 3: Flat Shapes All Around Us

**Unit Learning Goals**

- Students identify, describe, analyze, compare, and compose two-dimensional shapes.

This unit introduces students to the foundational concepts of geometry, with a focus on familiar flat (two-dimensional) shapes.

Students may initially associate names of shapes with everyday objects. For example, a rectangle is a shape that looks like a door. Students need to see and interact with many examples of a shape to accurately relate what’s in their environment to the geometric term.

For instance, students may say that only one of these two shapes is a triangle—the isosceles triangle sitting on its base—because they have seen examples like it being referred to as triangles. They may not consider a scalene triangle sitting on a vertex as a part of the same shape category because, in their experience, a shape like it hasn’t been associated with the term “triangle.”

Students explore differences in shapes and use informal language to describe, compare, and sort them. Circle, triangle, rectangle, and square are four shapes that students study and name here. (They will not describe what makes each shape so until grade 1.) Students also learn a key idea, that congruent shapes are still “the same” even if they are in different orientations.

Later in the unit, students use pattern blocks to make larger shapes. They reinforce their counting and comparison skills as they count and compare the pattern blocks used to create larger shapes. Students also use positional words (above, below, next to, beside) to describe the shapes they compose.

#### Section A: Exploring Shapes in Our Environment

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.3, K.CC.B, K.CC.B.5, K.G, K.G.A.1, K.G.A.2, K.G.B.4, K.G.B.5, K.MD.A.2, K.MD.B.3 |

**Section Learning Goals**

- Recognize and describe shapes in the environment.
- Use informal language to describe and compare shapes and their attributes.

In this section, students work to name, describe, and compare shapes in their environment more precisely. They focus on identifying circles, rectangles, squares, and triangles.

Students begin by identifying objects that look like flat shapes in books and in their surroundings. At this point, they are not yet expected to differentiate flat shapes from solid ones. For example, they may relate a tissue box to a rectangle. The difference between flat and solid shapes will be investigated in a later unit.

Likewise, students may not yet recognize distinctions in flat shapes with some similar features, such as a circle and an oval. Clarify that a shape is or is not as named, while acknowledging the connections students might be making. (“This shape is curved like a circle, but it is not a circle.”)

To help expand students’ mental image of shape categories, the shapes seen here are varied in size, type, and orientation.

When comparing shapes, students use their own language to describe how shapes are the same and different. They also consider the side length of rectangles and use “longer than” and “shorter than” to describe relative length. They learn that a square is a special kind of rectangle with all four sides having the same length (though are not required to know this definition).

PLC: Lesson 5, Activity 2, Triangle Sort

#### Section B: Making Shapes

**Standards Alignments**

Addressing | K.CC, K.CC.A.3, K.CC.B.4.c, K.CC.B.5, K.CC.C, K.CC.C.6, K.G, K.G.A.1, K.G.A.2, K.G.B.6 |

**Section Learning Goals**

- Explore shapes by putting shapes together to form larger shapes.

In this section, students develop spatial reasoning by manipulating shapes and solving geometric puzzles while using geometric language from earlier work.

Students use pattern blocks to compose geometric figures, explore shapes in different orientations, find shapes that match exactly, and complete puzzles that require reorienting shapes.

Throughout the section, students use their own language to describe how the shapes they are working with are alike and different, including descriptions of the side lengths of shapes in their comparison.

PLC: Lesson 13, Activity 2, Introduce Match Mine, Pattern Blocks

Estimated Days: 14 - 15

### Unit 4: Understanding Addition and Subtraction

**Unit Learning Goals**

- Students relate counting to addition and solve addition and subtraction story problems within 10.

In this unit, students develop their understanding of addition and subtraction as they represent and solve story problems.

Previously, students built their counting skills and represented quantities in a group with their fingers, objects, drawings, and numbers. Here, they relate counting to the result of two actions: putting objects together or taking objects away. Students enact addition by counting the total number of objects in two groups, and subtraction by counting what remains after some objects are taken away. (The word “total” is used here instead of “sum” to reduce potential confusion with the word “some” or part of a whole.)

Students then make sense of stories without questions and later solve story problems of two types—Add To, Result Unknown and Take From, Result Unknown. Students represent the problems in different ways, by acting them out, drawing, using numbers, or using objects. Connecting cubes should be accessible in all lessons for students who wish to use them, including for cool-downs. All story problems should be read aloud by the teacher, multiple times if needed.

Students are also introduced to expressions, a symbolic way to represent addition and subtraction. Initially, the teacher records the process of adding and subtracting using words such as “5 and 3” or “4 take away 1.” Later, students see that “5 and 3” and “4 take away 1” can be expressed by \(5+3\) and \(4-1\), respectively. They learn that these expressions are read “5 plus 3” and “4 minus 1.” (Students are not expected to read expressions out loud or to use precise language at this point.)

Later in the section, students connect expressions to pictures and story problems. They find the value of addition and subtraction expressions within 10.

In a future unit, students will compose and decompose numbers up to 10 and solve other types of addition and subtraction problems.

#### Section A: Count to Add and Subtract

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.B.5, K.OA.A.1 |

**Section Learning Goals**

- Understand addition as putting together and subtraction as taking from.

In this section, students learn to see adding as putting together two groups and counting the total number of objects, and subtracting as taking away a number of objects from a group and counting what remains.

They represent combining and removing with physical objects. No stories or contexts are used here so that students can focus on the actions of putting together, adding to, and taking from. The language “add,” “put together,” “subtract,” and “take away” is used throughout the section to describe addition and subtraction.

Students learn to interpret a phrase such as “5 and 3” to mean combining two groups (5 in one group and 3 in the other) and a phrase such as “5 take away 3” to mean finding what remains after removing 3 objects from a group of 5. They also hear language that describes the result of those actions, such as: “5 and 3 is 8” and “5 take away 3 is 2.” No symbolic notation is used at this point.

Students also encounter and count groups of images in scattered configurations for the first time. This task highlights the need to keep track of what has been counted.

To keep track of the dots in this example, students may count all the black dots first and then the white dots or cross off dots as they count. They may also count in no particular order. Students see that although they may count the dots in a different order, they arrive at the same total.

PLC: Lesson 3, Activity 2, Count Scattered Images

#### Section B: Represent and Solve Story Problems

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.3, K.CC.B.5, K.OA.A.1, K.OA.A.2 |

**Section Learning Goals**

- Represent and solve Add To, Result Unknown and Take From, Result Unknown story problems within 10.

In this section, students represent and solve story problems with playgrounds and parks as contexts. The types of problems are limited to Add To, Result Unknown and Take From, Result Unknown.

Students begin by acting out and representing stories that don’t include a question. Questionless story problems encourage students to think about the context and the action in the story without feeling pressure to solve the problem.

*There were 5 students jumping rope at recess.
2 more students came out to play with them.*

As questions are posed, students represent the problems with objects, math tools, drawings and numbers, and focus on explaining how their representation connects to the story. While they may represent a problem in any way that makes sense to them, students notice that organized drawings or objects make it easier to see the connections.

Students are also introduced to the concept of 0 representing a count of no objects. This idea may be abstract to students, so it is introduced in a Take From, Result Unknown story problem, where taking objects away leaves no remaining objects.

The term “expression” is introduced here. Students begin to see expressions as a way to record quantities being combined or removed. For instance, as a student describes what happens with their counters, the teacher writes the words “7 take away 3” and “\(7-3\),” and says “7 take away 3” and “7 minus 3.” Students are not expected to interpret expressions at this time.

PLC: Lesson 9, Activity 1, Markers at School

#### Section C: Addition and Subtraction Expressions

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B.4.c, K.OA.A.1, K.OA.A.2 |

**Section Learning Goals**

- Find the value of addition and subtraction expressions within 10.
- Relate addition and subtraction expressions to story problems.

In this section, students formally work with expressions for the first time. They match expressions such as \(3 + 2\) and \(8 - 1\) to story problems and drawings and articulate why an expression represents a given problem or drawing. While students fill in addition and subtraction expressions, they are not expected to produce expressions independently in this section.

\(5-3\)

\(2 +1\)

Students then transition from expressions that represent story problems or drawings to expressions without a context. To find the value of expressions, students may add or subtract in a way that makes sense to them, reasoning with fingers, objects, or drawings.

With repeated experience, students begin to notice regularity when adding and subtracting (MP8). For instance, they see that adding 1 results in the next number in the count sequence and that adding 0 results in the same number.

PLC: Lesson 17, Activity 2, Notice $+ 0$ and $+ 1$ Patterns

Estimated Days: 16 - 18

### Unit 5: Composing and Decomposing Numbers to 10

**Unit Learning Goals**

- Students compose and decompose numbers within 10.

In this unit, students explore different ways to compose and decompose numbers within 10 and to represent the compositions and decompositions.

Previously, students counted and compared groups and images of up to 10 objects. They solved addition and subtraction story problems and wrote expressions to represent the problems. Here, they use those experiences to compose and decompose numbers within 10. (The terms “make” or “break apart” are used with students.)

Special attention is given to composing and decomposing 10, as it is the basis of place value in our number system. To support their reasoning, students use their fingers and a 10-frame—created by putting together two 5-frames. They use these tools to think about pairs of numbers that make 10.

Symbolic notation develops slowly across the units. Students first complete expressions that represent numbers being composed and decomposed. In doing so, they also practice writing numbers without handwriting lines.

Later, students encounter equations of the form \(5 = 3 + 2\). Teachers read this equation as “5 is 3 plus 2.” Note that the equations are written with the total on the left side of the equal sign and the addends on the right. Aside from representing composition and decomposition, this notation helps students see that the equal sign means that “both sides have the same value,” rather than “the answer comes next.” In a later unit, students will see equations with the addends on the left side.

The work here prepares students to make sense of teen numbers in the next unit and lays the groundwork for students to develop fluency with addition and subtraction facts within 10 in grade 1. (For example, to find the sum of \(9 + 5\), they can decompose 5 into \(1 + 4\) and find \(9 + 1 + 4\) or \(10+ 4\).) Much of the addition and subtraction work in future grades also hinges on the idea of composing and decomposing numbers, 10 in particular.

#### Section A: Make and Break Apart Numbers to 9

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.2, K.OA.A.2, K.OA.A.3, K.OA.A.5 |

**Section Learning Goals**

- Compose and decompose numbers up to 9 in more than 1 way.
- Write expressions to represent decompositions.

In this section, students compose and decompose numbers to 9. They work with physical objects, such as counters and connecting cubes, that can help to show ways to make and break apart numbers.

As they progress through the lessons, students come to understand that there are different ways to compose and decompose a given number. They write expressions to record compositions and decompositions.

6 is 3 and 3

6 is 4 and 2

6 is 5 and 1

6 is 3 and 3

\(3 + 3\)

6 is 4 and 2

\(4 + 2\)

6 is 1 and 5

\(1 + 5\)

PLC: Lesson 2, Activity 2, Han’s Pattern Block Design

#### Section B: More Types of Story Problems

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.2, K.OA.A.1, K.OA.A.2, K.OA.A.3 |

**Section Learning Goals**

- Solve Put Together, Total Unknown, Put Together/Take Apart, Both Addends Unknown, Add To, Result Unknown, and Take From, Result Unknown story problems.

In this section, students represent and solve Put Together/Take Apart story problems—first where the total is unknown, and later where both addends are unknown. Students also see equations and learn the term for the first time.

*Jada made 6 paletas with her brother.
They made two flavors, lime and coconut.
How many of the paletas were lime?
Then how many of the paletas were coconut?*

Problems where both addends are unknown may be more challenging because there is no action in the story and more than one solution is possible. Students work to find multiple solutions but are not expected to find all the solutions in kindergarten.

To represent and solve story problems, students continue to use math tools and drawings, and to explain how their representation shows the story. They may use methods such as clearly separating the groups, using 2 colors, or using letter, word, and number labels to make their drawings easier for others to understand. Students also write expressions independently to record the solutions to the story problems.

Equations are introduced as a way to record the quantities and solutions in story problems. For instance, as a student explains a solution to the paleta problem, the teacher writes “\(6 = 2 + 4\)” and says: “Jada made 6 paletas, 2 in coconut flavor and 4 in lime flavor. We can write that as 6 is 2 plus 4.”

All equations in this unit are written with the total first (on the left side of the equal sign). Equations are read as “6 is 2 plus 4,” rather than “6 equals 2 plus 4.” Note that students are not expected to interpret equations at this time.

PLC: Lesson 8, Activity 1, Citrus Juice

#### Section C: Make and Break Apart 10

**Standards Alignments**

Addressing | K.CC, K.CC.A.3, K.CC.B, K.CC.B.5, K.OA.A.1, K.OA.A.2, K.OA.A.3, K.OA.A.4 |

**Section Learning Goals**

- For any number from 1 to 9, find the number that makes 10 when added to the given number.

This section focuses exclusively on composing and decomposing 10. This number is foundational to the understanding of place value and the work on numbers and operations in later grades.

Previously, students developed their understanding of the numbers 6–9 by relating it to 5 and using 5-frames. Here, students use a 10-frame—by putting together two 5-frames—and their fingers as tools to represent numbers and make and break apart 10 in different ways. The blank squares in the 10-frame and the fingers that are down allow students to see or count how many more are needed to make 10.

Throughout the section, students continue to build their familiarity with equations. They connect compositions and decompositions of 10 represented on their fingers and on 10-frames to addition equations and write missing numbers in such equations.

\(10 = 7 + 3\)

\(10 = 9 + 1\)

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

Students are not expected to write equations independently in kindergarten. And although students may start to learn combinations that make 10 from memory, fluency with sums of 10 is not required until grade 1.

PLC: Lesson 14, Activity 1, Introduce What’s Behind My Back?, 10 Cubes

Estimated Days: 13 - 15

### Unit 6: Numbers 0–20

**Unit Learning Goals**

- Students answer “how many” questions and count out groups within 20. They understand that numbers 11 to 19 are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. They write numbers within 20.

In this unit, students count and represent collections of objects and images within 20. They apply previously developed counting concepts—such as one-to-one correspondence, keeping track of what has been counted, and conservation of numbers—to larger numbers.

Previously, students have counted, composed, and decomposed numbers up to 10, using tools such as counters, connecting cubes, 5-frames, 10-frames, drawings, and their fingers. They wrote expressions to record compositions and decompositions.

Here, students use the 10-frame to organize groups of 11–19 objects and images. This tool encourages students to see teen numbers as 10 ones and some more ones, emphasizing the \(10+n\) structure of the numbers 11–19. They use this structure as they represent teen numbers with their fingers, objects, drawings, expressions, and equations. Students see equations with the addend written first, such as \(10 + 6 = 16\).

Throughout the unit, students practice tracing and writing numbers 11–20. It is common for students at this stage to write numbers backwards, so the emphasis is on writing a number that is recognizable to others. Reversing the order of the digits of teen numbers is also expected, due to how teen numbers are said in English. Repeatedly seeing the number 1 written first to represent teen numbers helps students recognize the structure of these numbers.

When tracing and writing numbers, students should write on a flat surface while sitting in a chair with feet flat on the floor. Number writing practice can also happen in other parts of the day and can be done using a variety of writing tools (crayons, colored pencils, markers, and so on) for increased engagement. Students can practice creating numbers with dough, tracing numbers in sand, or forming numbers with pipe cleaners.

#### Section A: Count Groups of 11-20 Objects

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B, K.CC.B.4, K.CC.B.4.a, K.CC.B.4.b, K.CC.B.5, K.OA.A.1, K.OA.A.2, K.OA.A.5 |

**Section Learning Goals**

- Count groups of up to 20 objects.

In this section, students count groups of 11–20 objects using strategies they developed earlier when working with smaller sets of objects.

Students participate in Counting Collections as the first activity in each lesson. They think about how organizing the objects can help ensure an accurate count and may use a counting mat or a 10-frame. Students also recognize that the number of objects in a group does not change, regardless of the way they are arranged.

Display written numbers for students whenever they share their count. In later sections, after seeing numbers displayed repeatedly, students will practice recognizing, tracing, and writing numbers 11–20. They will relate these numbers to addition expressions and equations. No expressions or equations are used in this section.

PLC: Lesson 4, Activity 2, Count, Rearrange, Recount

#### Section B: 10 Ones and Some More

**Standards Alignments**

Addressing | K.CC, K.CC.A.3, K.CC.B.4.a, K.CC.B.5, K.NBT.A.1, K.OA.A.1 |

**Section Learning Goals**

- Understand numbers 11-19 as 10 ones and some more ones.

In this section, students see the numbers 11–19 as 10 ones and some more ones. They compose and decompose teen numbers and record the compositions and decompositions with objects, drawings, and expressions.

Students use fingers and 10-frames to represent these numbers, but with more emphasis on the 10-frames as the lessons progress. To represent a teen number, they fill a 10-frame and show some more ones, which they may arrange in different ways. To determine the number of objects, students may count all or count on from 10 (though the latter is not an expectation in kindergarten).

Students compose and decompose teen numbers by starting with the parts (“10 and 5 is 15”) and starting with the total (“15 is 10 and 5”). For the first time, students see equations with the addends on the left side of the equal sign (\(10 + 5 = 15\)). They complete equations that show missing parts or a missing total to represent teen numbers as 10 ones and some more ones (\(\underline{\hspace{1cm}} + \underline{\hspace{1cm}} = 12\) and \(10 + 7 = \underline{\hspace{1cm}}\)).

Starting from this section, students have access to a reference sheet that shows numbers 11–20 with dots in 10-frames, which they can use to identify written numbers. Students can count the dots to determine which written number is on the card.

PLC: Lesson 8, Activity 2, Make Each Number

#### Section C: Count Groups of 11–20 Images

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B.4, K.CC.B.4.a, K.CC.B.4.b, K.CC.B.5, K.NBT.A.1, K.OA.A.1, K.OA.A.4 |

**Section Learning Goals**

- Count groups of images up to 20.
- Represent quantities up to 20 with a written number.

In this short section, students count groups of up to 20 images arranged in lines, arrays, circles, and on 10-frames.

Images arranged in a circle can be tricky to count, motivating a greater need to keep track of what has been counted. Students use their understanding that teen numbers are composed of 10 ones and some ones to help them count and keep track of groups of up to 20 images and then to write numbers to represent such quantities.

Throughout this section, students should have continued access to the reference sheet that shows numbers 11–20 with dots in 10-frames.

PLC: Lesson 12, Activity 2, Count in Circles

Estimated Days: 11 - 13

### Unit 7: Solid Shapes All Around Us

**Unit Learning Goals**

- Students identify, describe, analyze, compare, and compose two- and three- dimensional shapes. Counting, addition, and subtraction are revisited in the geometric contexts.

In this unit, students explore solid shapes while reinforcing their knowledge of counting, number writing and comparison, and flat shapes. They compose figures with pattern blocks and continue to count up to 20 objects, write and compare numbers, and solve story problems.

In an earlier unit, students investigated two-dimensional shapes. They named shapes (circle, triangle, rectangle, and square) and described the ways the shapes are different. Students used pattern blocks to build larger shapes and used positional words (above, below, next to, beside) along the way.

Here, students distinguish between flat and solid shapes before focusing on solid shapes. They consider the weight and capacity of solid objects and identify solid shapes around them.

Geoblocks, connecting cubes, and everyday objects are used throughout the unit. Standard geoblock sets do not include cylinders, spheres, and cones. When these shapes are required, “solid shapes” are indicated as required materials. If solid shapes are not available, students can work with everyday items that represent each shape.

Students use their own language to describe attributes of solid shapes as they identify, sort, compare, and build them, while also learning the names for cubes, cones, spheres, and cylinders.

3 cones

4 cubes

5 cylinders

*How many shapes did you use all together?*

The work here prepares students to identify defining attributes of shapes and to use flat and solid shapes to create composite shapes in grade 1.

#### Section A: Compose and Count with Flat Shapes

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.A.3, K.CC.B.5, K.CC.C, K.CC.C.6, K.CC.C.7, K.G.B.5, K.G.B.6, K.NBT.A.1, K.OA, K.OA.A.1, K.OA.A.2, K.OA.A.3, K.OA.A.4, K.OA.A.5 |

**Section Learning Goals**

- Compose shapes from smaller shapes.
- Count and compare numbers, and solve story problems involving shapes.

In this section, students strengthen their understanding of number concepts while working with pattern blocks. The work here allows the teacher to ensure that students have proficiency in counting and counting out to 20, writing and comparing numbers, and solving story problems.

In solving story problems, students match equations to the quantities in the problems, and complete equations so that they match the problems. For the first time, they hear equations read with the term “equals” rather than “is.” For example, \(9 - 6 = 3\) is read “9 minus 3 equals 6.” In this section, students see equations written with both the total written first and the addends written first.

Students consider ways to make the number 10 in the context of building shapes and completing puzzles with pattern blocks. Along the way, they think about attributes of pattern blocks.

\(4 + 6 = 10\)

\(10 = 6 + 4\)

\(2 + 8 = 10\)

\(10 = 8 + 2\)

PLC: Lesson 6, Activity 2, Many Ways to Make 10

#### Section B: Describe, Compare, and Create Solid Shapes

**Standards Alignments**

Addressing | K.CC.A.1, K.CC.B.5, K.G, K.G.A.1, K.G.A.2, K.G.A.3, K.G.B.4, K.G.B.5, K.G.B.6, K.MD.A, K.MD.A.1, K.MD.A.2, K.MD.B.3, K.OA.A.5 |

**Section Learning Goals**

- Compare weight and capacity of objects.
- Compose shapes from smaller shapes.
- Describe and compare three-dimensional shapes.

This section introduces students to solid shapes. Students begin by distinguishing solid shapes from flat shapes. They then learn about weight as an attribute of solid shapes and compare weights, and work with tactile materials or objects to develop their understanding of three-dimensional shapes.

Throughout the section, students hear and use the terms “flat” and “solid” to describe two- and three-dimensional shapes, but they also use their own language to talk about shapes. When comparing weights, the terms “heavy,” “light,” “heavier,” and “lighter” are used. While students are introduced to the names of solid shapes, they are not expected to use the formal terms. For example, they may say “ball” to refer to a sphere.

Initially, students build solid shapes with clay. Later, they do so out of given components, using positional words and names of shapes as they build and describe their creations. They also describe attributes of solid shapes as they compare and sort them.

At the end of the section, students create a model of their classroom and use solid shapes to represent objects in their world.

PLC: Lesson 11, Activity 1, Compare Solid Shapes

Estimated Days: 16

### Unit 8: Putting It All Together

**Unit Learning Goals**

- Students consolidate and solidify their understanding of various concepts and skills on major work of the grade. They also continue to work toward fluency goals of the grade.

In this unit, students revisit major work and fluency goals of the grade, applying their learning from the year.

Section A focuses on concepts of counting and comparing. Section B highlights the presence of math in students' school community. Section C enables students to practice composing and decomposing numbers within 5, as well as adding and subtracting within 5. Section D focuses on composing and decomposing 10.

The sections in this unit are standalone sections, not required to be completed in order. The goal is to offer ample opportunities for students to integrate the knowledge they have gained and to practice skills related to the expected fluencies of the grade.

The content here lays the foundation for grade 1, where students add and subtract fluently within 10 and count and compare larger quantities. Students will also learn about ten as a unit, which is the basis for understanding place value in the base-ten system.

#### Section A: Counting and Comparing

**Standards Alignments**

Addressing | K.CC, K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B.4, K.CC.B.4.c, K.CC.B.5, K.CC.C, K.MD.B.3, K.NBT.A.1, K.OA.A.2 |

**Section Learning Goals**

- Count and compare groups of objects and images.
- Represent and write numbers up to 20.

In this section, students count and compare collections of up to 20 objects. The focus is on the count sequence up to 20 and using it to determine 1 more or 1 less than a given quantity or number, both with and without a context.

*Compare the groups of objects.
Explain how you know which group has more objects.*

*There were 10 people on the bus. Then 1 person got off the bus.
How many people are on the bus now?
1, 2, 3, 4, 5, 6, 7, 8, *

**9**, 10

Many of the activities in this section are optional because the standards do not expect students to compare quantities or numbers greater than 10. This work prepares students to relate counting to addition and subtraction in grade 1.

PLC: Lesson 3, Activity 2, Singing Students

#### Section B: Math in Our School

**Standards Alignments**

Addressing | K.CC, K.CC.A, K.CC.A.3, K.CC.B, K.MD, K.OA, K.OA.A.1, K.OA.A.2, K.OA.A.5 |

**Section Learning Goals**

- Represent and write quantities and numbers up to 20.

In this section, students explore and describe the math around them. They participate in activities that allow them to notice, record, ask questions, and tell stories about math in their community.

First, students record quantities that they see in their school as they make their own number book. Next, they ask and answer their own mathematical questions, such as: “How many tiles are there from the office to the cafeteria?” or “Are there more doors or more windows in the library?"

*There are 5 pictures on one side of the hallway.
There are 3 pictures on the other side of the hallway.
How many pictures are there in the hallway?*

\(5+3=8\)

Finally, students create, share, and solve story problems about their environment and community. While the school building is used as a context, the activities in this section can be adapted for students to do in their home community.

PLC: Lesson 8, Activity 1, Find Someone Who

#### Section C: Fluency within 5

**Standards Alignments**

Addressing | K.CC.A.2, K.CC.C.6, K.MD.B.3, K.OA.A.2, K.OA.A.3, K.OA.A.5 |

**Section Learning Goals**

- Fluently add and subtract within 5.

In this section, students develop fluency with adding and subtracting within 5. Repeated practice with different compositions of numbers to 5 prepares students to fluently find the value of addition and subtraction expressions.

Students use a variety of tools and representations for their work with the numbers 1–5.

For instance, they sort domino cards based on the number of dots they have and sort addition and subtraction expressions by their value.

In the final lesson, students apply what they learned and use objects and equations to find a missing part with a total of up to 5.

\(5=3+2\)

\(1+4=5\)

\(5 - \underline{\hspace{.4 cm}3\hspace{.4 cm}} = 2\)

\(5 - \underline{\hspace{1 cm}} = 2\)

PLC: Lesson 13, Activity 2, Compare Dots on Dominoes

#### Section D: All About 10

**Standards Alignments**

Addressing | K.CC, K.CC.B.4, K.CC.B.4.c, K.CC.B.5, K.NBT.A.1, K.OA.A.1, K.OA.A.2, K.OA.A.3, K.OA.A.4 |

**Section Learning Goals**

- Use understanding of 10 to work with numbers to 20.

In this section, students work with 10 as a benchmark when working with numbers within 20. This work prepares them to add within 20 in grade 1, where students will be encouraged to make a ten.

The section begins with students composing and decomposing 10 in different ways and representing these compositions and decompositions with equations. Students then find the number that makes 10 when added to any given number. They also use their understanding of the magnitude of 10 to estimate if groups have more or fewer than 10 items.

Throughout the section, students use fingers, objects, drawings, 10-frames, and equations to represent their thinking. They also create a tool with 10 beads, 5 in each color, to show different compositions of 10.

Finally, students compose and decompose teen numbers 11-19, always working with a group of 10 ones and some more ones.

*“How many students will sit at the table? How many will sit on the rug?
How many students are there altogether?”*

\(\underline{\hspace{1.1 cm}}\)

\(+\)

\(\underline{\hspace{1.1 cm}}\)

\(= \phantom{la} \underline{\hspace{1.1 cm}}\)

PLC: Lesson 21, Activity 1, Where Will They Sit?

Estimated Days: 15 - 21