Kindergarten Mathematics Counting and Cardinality

K.CC.B.4

About the Math

Full Standard

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

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

Measurement Topic

This standard is reported on the report card in these quarters as follows:

**Kindergarten Students Learning Kindergarten Standards**
Quarter 1	Quarter 2	Quarter 3	Quarter 4
Report Card Measurement Topic: Demonstrates understanding of counting principles.

About the Math

Skills and concepts below are learning goals for this standard.

Count objects in a group (each object is counted only once) regardless of arrangement and order.
Determine "how many" are in a group after counting all the objects.
Count on from a known number (without recounting the whole group) when one more object is added to the group.
Explain how things are counted.
Explain that the next number is one more or one larger.

One of the first major concepts in a student’s mathematical development is cardinality. Cardinality is knowing that the number word said tells the quantity you have, and that the number you end on when counting represents the entire amount counted. The big idea is that number means amount and, no matter how you arrange and rearrange the items, the amount is the same. Until this concept is developed, counting is merely a routine procedure done when a number is needed. To determine if students have the cardinality rule, listen to their responses when you discuss counting tasks with them. For example, ask, “how many are here?”. The student counts correctly and says that there are seven. Then ask, “are there seven?”. Students may count or hesitate if they have not developed cardinality. Students with cardinality may emphasize the last count or explain that there are seven because they counted them. These students can now use counting to find a matching set.

Students develop the understanding of counting and cardinality from experience. Almost any activity or game that engages children in counting and comparing quantities, such as board games, will encourage the development of cardinality. Frequent opportunities to use and discuss counting as a means of solving problems relevant to kindergarteners is more beneficial than repeating the same routine day after day. For example, ask students questions that can be answered by counting up to 20 items before they change and as they change locations throughout the school building.

As students develop meaning for numerals, they also compare numerals to the quantities they represent. The models that can represent numbers, such as dot cards and dominoes, become tools for such comparisons. Students can concretely, pictorially or mentally look for similarities and differences in the representations of numbers. They begin to “see” the relationship of one more, one less, two more and two less, thus landing on the concept that successive numbers name quantities that are one larger. In order to encourage this idea, children need discussion and reflection of pairs of numbers from 1 to 10. Activities that utilize anchors of 5 and 10 are helpful in securing understanding of the relationships between numbers. This flexibility with numbers will build students’ ability to break numbers into parts.

Provide a variety of experiences in which students connect count words or number words to the numerals that represent the quantities. Students will arrive at an understanding of a number when they acquire cardinality and can connect a number with the numerals and the number word for the quantity they all represent.

Common Misconceptions

Some students might think that the count word used to tag an item is permanently connected to that item. So when the item is used again for counting and should be tagged with a different count word, the student uses the original count word. For example, a student counts four geometric figures: triangle, square, circle and rectangle with the count words: one, two, three, four. If these items are rearranged as rectangle, triangle, circle and square and counted, the student says these count words: four, one, three, two.

Progression of Standard within Kindergarten

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression Throughout Year
Quarter 1	Quarter 2	Quarter 3	Quarter 4
Count objects in a group (each object is counted only once) regardless of arrangement and order. Determine "how many" are in a group after counting all the objects. Count on from a known number (without recounting the whole group) when one more object is added to the group. Explain how things are counted.

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map. Information is not the complete standard.

Progression Across Grades
Grade 1
Relate counting to adding and subtracting (1.OA.5)

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Instructional Tasks

TASKS

These tasks can be used with small group or whole group instruction.

Using two different types of counters, have students grab a handful of the first counter and then match one of the second counters to each. You could use bear counters and cm cubes (base-ten unit blocks) and tell a story about each bear getting a package or present. Prompt students to count the number of first counters and the number of second counters. How do they know each one is matched to only one other object?
Line up a row of colored tiles in one color next to each other. Underneath make a row of the same amount of tiles in a different color but with more distance between each of the tiles. Discuss with students what they notice about how many we have of each. How can we tell if we have the same amount? Rebuild the two sets, but use 1 - 2 more tiles in the first set of tiles that are close together than the second set of tiles that are further apart. What do you notice now? How can we tell if we have the same amount?
Place two colored counters in a cup. Have students pour out all of the counters and count how many they have. Repeat with the same set. Ask students to count again. What was the same? What was different? Students can also work in pairs and take turns counting and recounting their sets. You could also have students use markers or paint dobbers to record the different ways they saw the counters.

SLIDE-BASED TASKS

These links are HCPSS created instructional tasks. These tasks are provided in Google slides. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

Module 2 • Counting to 10

Count Objects to 10 and Tell How Many:

Counting Leaves
Goodnight at the Zoo
Fireflies
Coconuts- Day 1
Hide the Numbers 1-10

Module 3 • Counting to 20

Count Objects to 20 and Tell How Many:

Count it Out
Dino Count

Additional Tasks

These links provide instructional ideas connected to this standard.

The Candyman (3 Act Task, G.Fletcher)
K.CC Goody Bags (Illustrative Mathematics)
K.CC Counting Mat (Illustrative Mathematics)
K.CC The Napping House (Illustrative Mathematics)
MSDE Lesson Seeds

Tasks Print Resources

These publications have been provided for the kindergarten team at each school. Check with your team leader if you cannot find them.

Print Resources
Book Thumbnail	Title	Reference
	Teaching Student-Centered Mathematics by: John A. Van de Walle and Lou Ann H. Lovin	p. 100
	Hands-On Standards published by: Learning Resources	Counting to 5 and Back, p. 16 Groups of 0-5, p. 18 Groups of 6-10, p. 20 Number Shapes, p. 22 Estimate and Count, p. 24 Comparing Groups, p. 26 Equal Groups, p. 28 More and Fewer, p. 30 Order of Numbers, p. 32 Ordinal Numbers, p. 34 Counting On, p. 36 Position of Objects, p. 52 Representing Numbers with Objects, p. 54
	Math Intervention K-2 by: Jennifer Taylor-Cox	Sleepy Bears, p. 18-19 Try to Go Home, p. 25-26 On the Right Track, p. 29-30 Rainbow of Beads, p. 33 Same or Not the Same, p. 36-37 Elephant Parade, p. 71-73 Clover Patch, 90-92
	Developing Number Concepts Book 1 by: Kathy Richardson	Slide and Check, p. 26 Creations, p. 33 Making Towers, p. 29 Break it Up, p. 43 Grow and Shrink, p. 35 Exploring Growing Patterns, p. 113 Creating Growing Patterns, p. 116
	Developing Number Concepts Book 2 by: Kathy Richardson	Number Arrangements, p. 78
	Basic Math Facts by Susan O'Connell and John SanGiovanni	I Spy One More, p. 44

Tasks Connected to Literature

Suggested titles to support the standard can be found in the table below. Check your school library or Howard County Library System for availability, or purchase using Materials of Instruction (MOI) funds. When available, select links to view activities aligned to each title.

One...Two...Three...Sassafras!

by Stuart J Murphy

1, 2, 3 Sassafrass (pdf)

Chicka Chicka 123

by Bill Martin Jr. and Michael Sampson

Every Buddy Counts

by Stuart J Murphy

Every Buddy Counts KCC (pdf)

How Many Bugs in a Box?

by David A. Carter

How Many, How Many, How Many

by Rick Walton

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Centers and Practice

Centers

These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

Count 1 to 10 Book (HCPSS-adapted resource)
Five Frame Memory (HCPSS-adapted resource)
Frame Flash (HCPSS-adapted resource)
Match the Numeral Five Frame (HCPSS-adapted resource)
Match the Numeral Ten Frame (HCPSS-adapted resource)
Roll and Fill (HCPSS-adapted resource)
Roll and Match Five Frame (HCPSS-adapted resource)
Spin and Match Ten Frame (HCPSS-adapted resource)
Ten Frame Memory (HCPSS-adapted resource)

PRACTICE/ASSESSMENT

These materials can be used for independent practice and/or for assessment purposes. These tasks align with the learning goals for the standard.

Assessment and Instructional Tasks

Assessment Labels

5 Exit Ticket Ideas for Quarter 1

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Assessment

Full Standard

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

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

Measurement Topic

The standard is reported on the report card through these measurement topics. Expand the measurement topic for a description of what students who meet expectation are able to do.

Kindergarten Students Learning Kindergarten Standards by Measurement Topic

Demonstrates understanding of counting principles

The student is able to...

Quarter 1

count using the correct number names in the standard order pairing each object with only one number name. (one-to-one correspondence)
count a set of objects and recognize that the last number said represents the total count of the set. (cardinality)
count a set, state the total amount and after the set is moved around (no additions or removals), states the total without recounting. (conservation)
count on one more after identifying "how many" without recounting the whole group.

KCC4 Anecdotal Data Collection Tool

Visit the SBIR (Standards Based Instruction and Reporting) page in Course Essentials for more information and clarification.

Rubric (for thinking and reasoning evidence)

Use this rubric for observation and constructed responses (tasks that require explanation, justification, and/or representation).

Rubric for Tasks (pdf)

Exemplars for this standard (if available)

These samples are examples of what it might look like for a student who MEETS EXPECTATIONS, is MAKING PROGRESS, and/or is MAKING LIMITED/NO PROGRESS.

TASK	MEETS EXPECTATIONS	MAKING PROGRESS	MAKING LIMITED/NO PROGRESS
Put out 9 cubes in a set and prompt the student to count and tell how many. Observe student behaviors. Repeat with 5 cubes and then 14 cubes.	Student counts all 3 sets correctly showing proficiency with one-to-one correspondence and cardinality consistently with only minor errors.	Student counts 1-2 sets correctly, but did not count all 3 sets correctly. Student attempts to use a strategy to count (putting them in a line, moving them one at a time, putting them in a ten frame), but is unable to show one-to-one correspondence and cardinality consistently.	Student points to the pile and says numbers. Student guesses how many there are. Student puts them in a line and counts, but cannot use one-to-one correspondence and cannot accurately state the total amount in any of the sets.

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