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

Students enter kindergarten with a range of counting experiences, concepts, and skills. So, this unit is designed to be accessible to all learners regardless of their prior experience. In the first three sections, activities do not require counting, 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 to answer, “how many?” questions. These questions reinforce the idea that counting is a way to tell how many objects there are. Counting up to 10 objects will support students in the next unit, which will focus more deeply on numbers 1–10.

The content of this unit is designed to establish the structures and routines for centers, to create norms for classroom learning, and to begin building a mathematical community. In the first section, lessons are shorter to give students time to learn these routines and norms and to develop a mathematical community.

At different points throughout the unit, consider asking individual students to count a small group of objects. As the student works, observe the skills or understandings in the Checklist provided at the beginning of each section and in the Unit 1 Sections A–D Checkpoint document in the teacher resource packet. 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 students who have demonstrated the skills on the checklist throughout the unit.

In this unit, students continue to develop counting concepts and skills, including comparing groups of objects and images, and representing quantities with objects, pictures, and numbers.

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

Here, students rely on familiar activity structures to build counting skills and an understanding of the connection between quantities and numbers. Students first count groups of objects. Then they count groups of images. Objects and images appear in different arrangements, such as lines, arrays, number cube patterns, and on 5-frames. This helps build an understanding that changing the arrangement doesn’t change the quantity.

Use of fingers and 5-frames to represent numbers are emphasized to help students see the structure of numbers 6–10 as \(5+n\). Fingers are also always available and help with counting.

These fingers show 3.

These fingers show 8.

Students also compare numbers of objects and images. To describe the comparisons, students start by using the terms “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 they will develop proficiency with the distinction over time.

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 objects in their environment to the geometric term.

For instance, students may say that only one of the two shapes is a triangle—the isosceles triangle sitting on its base—because they have seen examples like it 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 defines each shape 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.

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

Previously, students developed their counting skills. Students learn addition as an extension of counting though joining two groups and counting to find the total. Students also extend their counting through subtraction. They count to find and remove objects within a collection and then count what remains. (The word “total” is used instead of “sum” to avoid confusion with the word “some” or part of a whole.)

Students then represent and solve Add To, Result Unknown and Take From, Result Unknown story problems. Students represent the problems in different ways, by acting them out, drawing, using numbers, or using objects. Connecting cubes and two-color counters should be made accessible in all lesson activities, including cool-downs, for students that want to use them throughout the unit.

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 these expressions are read as “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 unit, 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.

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. Students solved addition and subtraction story problems and wrote expressions to represent the problems. In this unit, students 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. 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 in this unit 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.

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 counted, composed, and decomposed numbers up to 10. They used counters, connecting cubes, 5-frames, 10-frames, drawings, their fingers, and other tools. They also 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 and some more, emphasizing the \(10+n\) structure of the numbers 11–19. Students use this structure as they represent teen numbers with their fingers, objects, drawings, expressions, and equations. Students see equations with the addends written first, such as \(10 + 6 = 16\). It is important to note that students are not expected to think of 10 ones as a unit called “a ten” or refer to single units as "ones" until Grade 1.

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.

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 weights and capacities 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.

The mathematical names cube, cone, sphere, and cylinder are introduced in this unit; however, students are not expected to use the names of solid shapes. Students can and are encouraged to continue to use their own language to describe and identify solid shapes.

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.

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

Section A focuses on the 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.
 

\(10 = \underline{\hspace{.4 cm} 8 \hspace{.4 cm}} + \underline{\hspace{.4 cm} 2 \hspace{.4 cm}}\)

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