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

### Cluster

Count to tell the number of objects.

Evidence of Learning Statements

 Students with a level 1 understanding of this standard will most likely be able to: Students with a level 2 understanding of this standard will most likely be able to: Students with a level 3 understanding of this standard will most likely be able to: Students with a level 4 understanding of this standard will most likely be able to: Begin counting. Students are emergent counters at this level. When counting objects, students may say number words and make an attempt at one-to-one correspondence. They may double count, miss objects, or the number sequence may be out of order. Say the number names in the standard order, using one-to-one correspondence when counting 10 objects. Inconsistently demonstrate that they understand that the last number name said tells the numbers of objects counted and inconsistently identify that the number of objects is the same regardless of their arrangement or the order in which they were counted. Inconsistently demonstrate that they understand each successive number name refers to a quantity that is one greater. Say the number names in the standard order, using one-to-one correspondence for numbers when counting more than objects.  Students can typically easily work with the teen numbers. Demonstrate that they understand that the last number name said tells the numbers of objects counted and identify that the number of objects is the same regardless of their arrangement or the order in which they were counted. Demonstrate that they understand each successive number name refers to a quantity that is one greater. Quickly recognize and name (subitize) how many objects are in a group without counting for multiple representations of the same number. Demonstrate that they understand when a counter is added or removed from a set that the count is one more/one less than the previous count without recounting the set.

Instructional Focus Statements

Level 3:

Instruction should move away from rote counting so that students are connecting number names with the number of objects that number represents in a set. Cardinality refers to the actual count or number of items in a set. Students should begin counting physical objects in order to develop a conceptual understanding of cardinality. In order to count a set of objects, students pair each word said with one object. This is usually facilitated by an indicating act such as moving each object keeping each word said paired with one and only one object. This helps develop an understanding of one-to-one correspondence.

Students need to develop an understanding that the last number name said when counting objects in a set tells the number of objects counted. Prior to reaching this conceptual understanding, a student who is asked “How many blocks?” may regard the counting process itself as the answer, as opposed to the number corresponding to the final object in the set. Students should be allowed to experience counting and discuss what happens when the same number of objects are arranged in differing ways allowing them to discover the second part of standard K.CC.4b—that the number of objects is the same regardless of their arrangement or the order in which they were counted.

Finally, students develop an understanding that each successive number name refers to a quantity that is one larger. As students are developing this understanding, they may have to entirely recount a set of known cardinality when an object is added to the set to indicate that it is one larger. It is important that students gain this understanding as it is a conceptual start for the addition strategy of counting on in grade 1.

Ultimately, throughout the counting and cardinality standards, it is important that students connect physical objects, oral number words, and the printed numerals.

Level 4:

As students solidify their understanding of cardinality, it is important that they develop the skill of quickly being able to recognize the number of objects in a group (subitizing). This skill will be very beneficial to students in subsequent grades. It allows them to develop strong number sense strategies such as unitizing, counting on, composing numbers, and decomposing numbers. Additionally, students should be challenged to think about what happens when an object is removed from a set. How does this effect the cardinality? Ultimately the more opportunities students have to work with varying representations of numbers, the more prepared they will be for future work.