Scope and Sequence
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: Matemáticas en nuestro mundoUnit 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: Exploremos nuestras herramientasStandards Alignments
|Addressing||K.CC, K.G, K.G.B|
- 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, Observa y pregúntate: Fichas geométricas
Section B: Reconozcamos cantidadesStandards Alignments
- 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.”
Students are introduced to two new routines, Act It Out and How Many Do You See, and the Picture Book center, where they look for small groups of images in picture books and create their own picture books by drawing pictures of groups of things they see in their world.
This section provides continued opportunity to formatively assess students’ counting concepts and skills.
PLC: Lesson 9, Activity 3, Conozcamos “Libros de imágenes: Crea”
Section C: ¿Hay suficientes?Standards Alignments
- 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 3, ¿Hay suficientes?
Section D: Contemos coleccionesStandards Alignments
|Addressing||K.CC, K.CC.A.1, K.CC.B, K.CC.B.4, K.CC.B.4.a, K.G.B|
- 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 2, Contemos colecciones
Estimated Days: 16 - 17
Unit 2: Números del 1 al 10Unit 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: Contemos y comparemos grupos de objetosStandards 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|
- 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.)
A few centers that focus on counting and comparing are introduced here: Math Fingers, Shake and Spill, and Less, Same, More. Students also begin tracing numbers in the Number Race center.
Shake and Spill
Less, Same, More
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 3, Conozcamos “Revuelve y saca: ¿Cuál es más?”
Section B: Contemos y comparemos grupos de imágenesStandards Alignments
|Addressing||K.CC.B, K.CC.B.4, K.CC.B.4.b, K.CC.B.5, K.CC.C.6|
- 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 and learn about a new center, Math Stories.
“Are there enough cartons of milk for each student? How do you know?”
PLC: Lesson 9, Activity 3, Más y menos con tableros de 5 y dedos
Section C: Conectemos cantidades y númerosStandards Alignments
|Addressing||K.CC, K.CC.A.3, K.CC.B.4, K.CC.B.5, K.CC.C.6|
- 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.
“Draw a line from each number to the group of dots that it matches.”
In many activities, students have opportunities to look for and make use of structure to help them with the tasks at hand (MP7).
Two new centers, Math Libs and Bingo, are introduced in this section.
PLC: Lesson 14, Activity 2, Los ingredientes de la pizza
Section D: Comparemos númerosStandards 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|
- 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, Actuémoslo: Tenedores para la cena
Estimated Days: 21 - 22