**Math: **Counting and Cardinality

**CC.1**

**Anchor Standard**

Count to 100 by ones and by tens.

__Know number names and the count sequence__

Students rote count by starting at one and counting to 100. When students count by tens they are only expected to master counting on the decade (0, 10, 20, 30, 40 …). This objective does not require recognition of numerals. It is focused on the rote number sequence.

*** Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: number words

**(zero - one hundred)**

**CC.2**

**Anchor Standard**

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

__Know number names and the count sequence__

Students begin a rote forward counting sequence from a number other than 1. Thus, given the number 4, the student would count, “4, 5, 6, 7 …” This objective does not require recognition of numerals. It is focused on the rote number sequence 0-100.

**CC.3**

**Anchor Standard**

Write numbers from

__0 to 20__. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

__Know number names and the count sequence__

Students write the numerals 0-20 and use the written numerals 0-20 to represent the amount within a set. For example, if the student has counted 9 objects, then the written numeral “9” is recorded. Students can record the quantity of a set by selecting a number card/tile (numeral recognition) or writing the numeral. Students can also create a set of objects based on the numeral presented. For example, if a student picks up the number card “13”, the student then creates a pile of 13 counters. While children may experiment with writing numbers beyond 20, this standard places emphasis on numbers 0-20.

Due to varied development of fine motor and visual development,

**reversal of numerals**is anticipated. While reversals should be pointed out to students and correct formation modeled in instruction, the emphasis of this standard is on the use of numerals to represent quantities rather than the correct handwriting formation of the actual numeral itself.

**CC.4**

**Anchor Standard**

Understand the relationship between numbers and quantities; connect counting to cardinality.

__Know number names and the count sequence__

Students count a set of objects and see sets and numerals in relationship to one another. These connections are higher-level skills that require students to analyze, reason about, and explain relationships between numbers and sets of objects. The expectation is that students are comfortable with these skills with the numbers 1-20 by the end of Kindergarten.

**a**. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

Students implement correct counting procedures by pointing to one object at a time (one-to-one correspondence), using one counting word for every object (synchrony/ one-to-one tagging), while keeping track of objects that have and have not been counted. This is the foundation of counting.

**b.**Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

Students answer the question “How many are there?” by counting objects in a set and understanding that the last number stated when counting a set (…8, 9,

**10**) represents the total amount of objects: “There are

**10**bears in this pile.”

**Since an important goal for children is to count with meaning, it is important to have children answer the question, “How many do you have?” after they count. Often times, children who have not developed cardinality will count the amount again, not realizing that the**

__(cardinality).__**10**they stated means 10 objects in all.

Young children believe what they see. Therefore, they may believe that a pile of cubes that they counted may be more if spread apart in a line. As children move towards the developmental milestone of conservation of number, they develop the understanding that the number of objects does not change when the objects are moved, rearranged, or hidden. Children need many different experiences with counting objects, as well as maturation, before they can reach this developmental milestone.

c. Understand that each successive number name refers to a quantity that is one larger.

Another important milestone in counting is inclusion (aka hierarchal inclusion).

**is based on the understanding that numbers build by exactly one each time and that they nest within each other by this amount. For example, a set of three objects is nested within a set of 4 objects; within this same set of 4 objects is also a set of two objects and a set of one. Using this understanding, if a student has four objects and wants to have 5 objects, the student is able to add one more- knowing that four is within, or a sub-part of, 5 (rather than removing all 4 objects and starting over to make a new set of 5). This concept is critical for the later development of part/whole relationships.**

__Inclusion__Students are asked to understand this concept with and without (0-20) objects.

For example, after counting a set of 8 objects, students answer the question, “How many would there be if we added one more object?”; and answer a similar question when not using objects, by asking hypothetically, “What if we have 5 cubes and added one more.

How many cubes would there be then?”

*******

**Count to tell the number of objects.**

Students use numbers, including written numerals, to represent quantities and to solve quantitative problems such as counting objects in a set, counting out a given number of objects, and comparing sets or numerals.

When learning to count, it is important for students to connect the

__collection of items__

(4 cubes), the

__number word__(“four”), and the

__numeral__(4), ultimately creating a mental picture of a number. If students simply rote-count a collection of objects without connecting these three components together, they “engage in a meaningless exercise of calling numbers that are one more than the last.”

**is the ability to “instantly see how many,” it helps students form a mental picture of a number. When students recognize a small collection of objects (e.g., 2 sets of two dots) as one group (e.g., four) – they are beginning to unitize. This ability to see a set of objects as a group is an important step toward being able to see smaller groups of objects within a total collection- which is necessary to decompose numbers. Materials such as dot cards, dice, and dominoes provide students opportunities to see a variety of patterned arrangements to develop instant recognition of small amounts.**

__Subitizing__**CC.5**

**Anchor Standard**

Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.

__Know number names and the count sequence__

In order to answer “how many?” students need to keep track of objects when counting. Keeping track is a method of counting that is used to count each item once and only once when determining how many. After numerous experiences with counting objects, along with the developmental understanding that a group of objects counted multiple times will remain the same amount, students recognize the need for keeping track in order to accurately determine “how many”. Depending on the amount of objects to be counted, and the students’ confidence with counting a set of objects, students may move the objects as they count each, point to each object as counted, look without touching when counting, or use a combination of these strategies. It is important that children develop a strategy that makes sense to them based on the realization that keeping track is important in order to get an accurate count, as opposed to following a rule, such as “Line them all up before you count”, in order to get the right answer.

As children learn to count accurately, they may count a set correctly one time, but not another. Other times they may be able to keep track up to a certain amount, but then lose track from then on. Some arrangements, such as a line or rectangular array, are easier for them to get the correct answer but may limit their flexibility with developing meaningful tracking strategies, so providing multiple arrangements help children learn how to keep track. Since scattered arrangements are the most challenging for students, this standard specifies that students only count up to 10 objects in a scattered arrangement and count up to 20 objects in a line, rectangular array, or circle.

Higher level thinkers count groups of objects by sets. For example: A set of 20 cubes might be counted by 2’s, 4’s, 5’s and 10’s.

**CC.6**

**Anchor Standard**

Identify whether the number of objects in one group is

**the number of objects in another group, e.g., by using matching and counting strategies.**

__greater than, less than, or equal to__Include groups with up to ten objects.

__Know number names and the count sequence__

Students use their counting ability to compare sets of objects (0-10). They may use matching strategies (Student 1), counting strategies (Student 2) or equal shares (Student 3) to determine whether one group is greater than, less than, or equal to the number of objects in another group.

__Student 1__I lined up one square and one triangle. Since there is one extra triangle, there are more triangles than squares. (Matching Strategy)

__Student 2__I counted the squares and I got 4. Then I counted the triangles and got 5. Since

5 is bigger than 4, there are more triangles than squares. (Counting Strategy)

__Student 3__I put them in a pile. I then took away objects. Every time I took a square, I also took a triangle. When I had taken almost all of the shapes away, there was still a triangle left.

That means that there are more triangles than squares. (Equal Shares)

**CC.7**

**Anchor Standard**

__Compare two numbers__between 1 and 10 presented as written numerals.

__Know number names and the count sequence__

Students apply their understanding of numerals 1-10 to compare one numeral from another. Thus, looking at the numerals 8 and 10, a student is able to recognize that the numeral 10 represents a larger amount than the numeral 8. Students need ample experiences with actual sets of objects before completing this standard with only numerals.