Students use an indicating act like moving objects or pointing to them to pair each number word said with one object only. Counting objects in a line is easiest, and students later keep track of counting objects in more difficult arrangements.

Students connect counting to cardinality by understanding that the last number said in a counting sequence indicates the number of objects in the counted set.

Students initially consider the act of counting as the answer to a “How many ___?” question. With more experience students realize that the number of objects counted remains the same even if rearranged or if the objects are counted in a different order.

Student uses the standard counting order when counting the animals.

Student touches and counts each animal only one time and says only one number when touching each animal.

Student moves animals or otherwise keeps track of which animals have been counted.

Student tells how many are in the set of animals by stating the last number counted.

How can you be sure you counted all of the animals?

How do you know that ___ is the total number of animals?

What strategy did you use to count the animals?

The numerals 1, 2, 3,…, 9 and 0 are arbitrary markers for the first numbers in the counting sequence. Students learn that these numerals are used in different ways and in patterns in larger numbers.  

Students explore the “teen” numbers, 11 through 19, through manipulatives, drawings, and equations to realize that these numbers represent ten ones and some more ones. Students group objects in the teen numbers to see the group of ten and the additional ones. 

The teen numbers present challenges for children because they do not clearly indicate their base-ten meanings. “Eleven” and “twelve” do not sound like “ten and one” and “ten and two.” In the remaining teen numbers, the word “teen” must be interpreted as meaning “ten”, but the order is reversed because the number of ones is stated before the ten, as in “sixteen” meaning six ones and ten. Additionally, students may initially read 16 as “one, six” before they come to understand the meaning of the 1 as ten.

Students count all numbers in the sequence without skipping or repeating numbers

Students count all objects by moving, grouping, or touching without skipping or repeating objects

Number reversals are developmentally appropriate and may require handwriting practice.

Are you sure that you counted all of the objects?  How could you be sure?

What number comes before ____?  

What number comes after _____? 

Initially, students may use visual cues to decide which set is larger. They eventually learn to count or match objects in the sets to determine and compare the sizes of the sets. Students compare sets of objects by matching objects in each set and looking for any extra in one set to indicate that that set is larger than the other.

Students also use their knowledge of the counting sequence (ie. which number is farther along in the sequence) to determine which set is larger.

As students match objects in sets, they progress to understanding that objects matched in each set represent the same number in each set.

Student builds a set with fewer than a given number.

Student builds a set with more than a given number.  

Student justifies the number of counters in each set using a counting and matching method and/or their knowledge of the count sequence.

Student uses appropriate terminology (more, fewer, the same) to describe and compare the sets. 

How do you know this set has fewer than 7 hearts?

How do you know this set has more than 7 hearts?

Which set has the most hearts? How do you know?

Which set has the fewest hearts? How do you know?

Initially students may use visual cues to decide which set is greatest or least. They eventually learn to count or match objects in the sets to determine and compare the sizes of the sets. Students compare sets of objects by matching objects in each set and looking for any extra in one set to indicate that that set is larger than the other.

Students also use their knowledge of the counting sequence (ie. which number is farther along in the sequence) to determine which set is greater.

As students match objects in sets, they progress to understanding that objects matched in each set represent the same number in each set.

Students will use a method to count and match up the objects in each set.

Students will justify the order of their sets of objects using their counting and matching method and/or their knowledge of the count sequence.

Student will use appropriate terminology (least, greatest) to describe the ordered sets of objects.

Why did you put the sets of fish in this order?

Why does this set of fish come before/after this set?

How can you describe the order of the sets of fish?

When ordering from greatest to least/least to greatest, how did you know which number to put first?

Students progress from initially chanting and stringing together multiple number words, to articulating each individual number with distinct words. This indicates their understanding of the uniqueness of each spoken number. In particular, they progress in articulating similar-sounding numbers, such as “fourteen” and “forty.”

Students learn that teen numbers name the number of ones before “teen,” meaning 10, unlike the numbers in the decades from 20 to 100 in which there is agreement between the written and spoken word, with the number of tens written and spoken first. Students develop an understanding that the decade words indicate the number of tens, such as 2 tens (20), 3 tens (30), 4 tens (40), etc. and that the suffix “-ty” means ten.

Students initially make mistakes such as “twenty-nine, twenty-ten, twenty-eleven, twenty-twelve…” until they develop an understanding of the relationship between written and spoken number words. They eventually learn to make decade transitions, and they come to recognize that the decade sequence mirrors the single-digit sequence.

Student says each number in sequence from 0 to 100.

Student differentiates between similar sounding numbers, such as fourteen/forty and fifteen/fifty.

Student applies patterns in their count sequence. 

Student transitions from one decade to the next (nineteen, twenty, twenty-one… twenty-nine, thirty, thirty-one… thirty-nine, forty, forty-one). 

What patterns do you notice when counting to 100? 

What do you notice whenever you get to a number that ends in 9 (i.,e., 9, 19, 29, 39 etc.)? Why does the next number you say end in zero (i.e., 10, 20, 30, etc.)?  

Numbers follow predictable patterns. You can count up and you can count down from any given number.

Humans have 10 fingers. These can be used as an aid no matter the circumstance.

0 comes before 1. 

Rote counting sets a base for subtraction.

Student says 0 after they say 1.

Each number is said in sequence.

What number comes before 1? What does 0 mean?

Can you count backward from 3? What about 5?

Why is it important to count backward?

Students can count by ones through 100, including the decade transitions from 39 to 40, 49 to 50, and so on, starting at any number. (Clements)

Students recognize that each counting number identifies a quantity that is one more than the previous number. (van de Walle, pg. 128) 

Students begin to count on, counting verbally and with objects from numbers other than 1. Students can determine immediately the number just before or just after another number without having to start back at 1. (Clements) 

Student uses patterns and structure in numbers to identify the number before a given number. 

Student uses patterns and structure in numbers to identify the number after a given number. 

Student provides reasoning for why a number comes after.

Student provides reasoning for why a number comes before.

What comes before 7? How do you know?

Why does 6 come before 7?

What comes after 69? How do you know?

Why do we say 70 after 69?

Student “sees” groups of ten within a quantity and counts those groups by 10 (relating to algebraic thinking and multiplication) (Clements, D. H., & Sarama, J. (2019)).

Understanding 10 as an important building block of large numbers sets students up for success in understanding place value.

Having one-to-one correspondence is crucial to being able to skip count. Students must trust that each group holds the amount they’re skip counting by, and this needs to be able to be confirmed.

When creating groups of 10, student has strong one-to-one correspondence.

When creating groups, once student gets to 10, they start over with a new group of 10 instead of saying “11, 12,...” etc.

When skip counting, student touches each group and counts in order, not skipping any decade number.

How many counters are in this group? (Point to a random group, answer should always be 10). How do you know?

When you count by 10s, how do you know what number comes next?

Why is it important to know how to skip count?

What skip count comes after 100 when counting by 10?

Students come to understand that a whole is larger than its parts (ie. 5 is larger than 2 and 3).

Students understand that numbers are nested inside of each other when counting and the next number is one more (ex. 9 is nested inside of 10, 9+1 = 10). (Clements, D.H., & Sarama, J. 2009) 

Students learn to identify the size of a whole by composing its smaller parts. They develop intuitive understandings of commutativity (ie. 2 and 3 is the same as 3 and 2) and associativity (ie. 1 and 1 and 3 is the same as 2 and 3 and 1 and 4).

Students subitize, or visually recognize the number of items in a collection, using spatial and numeric structures, such as common dot patterns (ie. 2x2 array).

Student uses their knowledge of visual patterns (subitizes) to identify an unknown part of a set up to 5.

Student uses a counting strategy to determine the unknown part of a set up to 5.

What are all the ways you can make 5? Which combination could this be? How do you know?

How can you figure out the unknown part?

How can you show (with drawings, manipulatives, tens frame, number beads/board, etc.) this part-whole relationship?

Students build on their subitizing skills and visual recognition of numbers up to 5 to learn combinations up to 10.

Students first learn familiar combinations, such as common doubles facts (ie. 5 and 5).

The use of a ten-frame structure supports students’ recognition of combinations up to 10 and beyond.

Student uses their knowledge of visual patterns (subitize) to identify an unknown part of a set of up to 10.

Student uses a counting strategy to determine the unknown part of a set up to 10.

What combinations make 9? Which combination could this be? Why?

How can you figure out the unknown part?

How can you show (with drawings, manipulatives, tens frame, rekenrek, etc.) this part-whole relationship?

Teachers should be using language like “half” and “halves” to describe two equal shares of an object or set.

Fair shares are equal. Equal means the same.

Shapes can be proportioned in different ways. You can divide a rectangle horizontally OR vertically and still maintain halves.

Use concrete representation if necessary.

Equal parts are the same.

Describe the equal part as “one half”.

Are these pieces the same size? How do you know?

How much of the whole brownie do we each get?

(When using the set of “cookies”) What is a fair way to make sure we both get the same number of cookies?

Students use objects, fingers, and math drawings to act out addition and subtraction situations, allowing them to “mathematize the real world” (Common Core Progression, 2019, p. 17).

Students need experience acting out problems of each different addition and subtraction type, including join, separate, part-part-whole contexts.

Students should be presented with problems in picture and word format.

Student uses standard counting order when counting objects.

Student uses a counting strategy to count each object only one time and says only one number when counting each object.

Student solution aligns to the numbers and action they used.

Student describes, draws, or models the action in the problem.

How can you be sure you counted all of the animals?

How can you be sure you counted each animal only once?

How do you know that ___ is the total number of animals?

How can you show what happened in the problem?

The value of a coin changes what it is worth. One dime has the value of ten cents. It is the equivalent of ten pennies.

The value of a coin can make it worth more than another coin. One dime is worth more than one nickel. 

Coins have different attributes. That’s how you can tell them apart.

Pennies are easily differentiated from the rest of the coins.

Both nickels and quarters look like “big” coins. Notice how student differentiates them based on other characteristics.

Student identifies the coin and creates a correct collection of pennies to match the value of the coin

What clues tell you that that coin is a (penny/nickel/dime/quarter)?

How can you tell the difference between (nickel & quarter/ penny & dime)?

A calendar organizes time into days, weeks, months, and years.

Time never stops.

Calendars have patterns that follow rules.

Student is appropriately differentiating between days and months.

Student is appropriately differentiating between yesterday, today, and tomorrow.

Is “yesterday” about the past, present, or the future?

What day of the week is after Saturday?

Can you always tell what tomorrow will be if you know what day it is today? How?

This standard is heavy in vocabulary. Be especially mindful of students with limited vocabulary as you teach this standard. Use explicit vocabulary teaching strategies, giving students hands-on experiences with each term. Explicit definitions can be found in the Curriculum Framework.

Volume, length, height, and weight are attributes that can be separated from size (big/small).

There are strategies and tools used to compare objects to one another.

Physically aligns two objects to determine which is longer or if they are the same length.

Student is using appropriate vocabulary to describe the comparison (The cow is heavier than the milk carton).

Comparing time may be challenging for students, as it is an abstract concept. Student may act out the events in order to make a conjecture about which is longer or shorter.

How were you able to tell that one object is (longer/shorter/heavier/lighter/more/less)?

What do you know about that object that helps you compare it?

Can you act out the event to try to figure out how to compare?

Presentation of triangles, rectangles, and squares should be made in a variety of spatial orientations so that students are less likely to develop common misconception that triangles, rectangles, and squares must have one side parallel to the bottom of the page on which they are printed.

Children should have experiences with different types of triangles (e.g., equilateral, isosceles, scalene, right, acute, obtuse); however, at this level, they are not expected to name the various types. 

A common misconception students have when a figure such as a square is rotated is they will frequently refer to the rotated square as a diamond. Clarification needs to be ongoing (e.g., a square is a square regardless of its location in space; there is no plane figure called a diamond).

This standard is heavy in vocabulary. Be especially mindful of students with limited vocabulary as you teach this standard. Use explicit vocabulary teaching strategies, giving students hands-on experiences with each term. Explicit definitions can be found in the Curriculum Framework.

Squares and rectangles are differentiated.

What clues tell you that shape is a (square/rectangle/triangle/circle)?

Do (squares/triangles) always have a side that sits on the bottom?

What’s another way to describe where the shape is?

Data are pieces of information collected about people or things. The primary purpose of collecting data is to answer questions. The primary purpose of interpreting data is to inform decisions.

When data are presented in an organized manner, students can interpret and discuss the results and implications of their investigation.

Data is collected all around us. Draw attention to the lunch count, the weather graph, or the “How Many Days in School?” chart for a glimpse into data.

Data can be represented in multiple ways, yet draw the same conclusion (bar graph vs. picture graph).

The first image in the graph is the anchor, NOT a part of the data collection. Children may count it in their data set.

Student may confuse the words “greatest” and “least”.

Students may count all of the data points, not just the ones you’ve asked them too. 

How do you know that category has the (greatest/least) votes?

How many (more/fewer) votes does ___ have than ___?

What do you think is the best way to show data?

Children intuitively recognize similarities between objects or situations, leading to the ability to group similar and dissimilar objects. 

Students may begin by grouping objects by general resemblance, and later by formal attributes (size, color, shape, etc.) but may switch attributes during a sort.

Students learn to follow verbal instructions for sorting objects and fixing a sorted set. Early on they make mistakes in these sorting activities, but eventually they consistently and accurately sort by a single attribute and re-classify a sorted set by a different attribute.

Student can articulate their rationale for sorting the objects.

Student classifies the objects into piles based on a single attribute (ie. color OR shape OR size).

Student name the attribute used to sort the objects. 

Why did you put these shapes in a set together?

What do these shapes have in common?

How do you know where this shape should go?

What other ways might you sort these shapes?

Patterning is a fundamental cornerstone of mathematics, particularly algebra. The process of generalization leads to the foundation of algebraic reasoning.

Eventually, students learn to find patterns in numbers, such as skip counting, which leads to understanding multiplication and division.

A variety of patterns are introduced. Examples include:

• ABABABAB

• ABCABC

• ABBAABBA

• AABBAABBAABB

• AABAAB.

Translates patterns into new media or using new materials; that is, abstract and generalize the pattern.

Student extends the pattern by two repetitions of the core, not just by a few shapes.

Student translates the given pattern using two different manipulatives/shapes/colors/numbers/etc.

Student is able to create a repeating (not growing!) pattern.

How can you identify the core of the pattern?

How is your pattern like the pattern you see above?

What is the core of your pattern? How many times will it repeat?