3-Step 9 Draft P-12 Mathematics Standards PDF (1).pdf

June 2025

Draft P-12 Utah Core Mathematics Standards

Released for Public Review by the Utah State Board of Education

Draft P-12 Utah Core Mathematics Standards Table of Contents

Introduction

Standards: Preschool Age 3

Standards: Preschool Age 4

Standards: Kindergarten

Standards: Grade 1

Standards: Grade 2

Standards: Grade 3

Standards: Grade 4

Standards: Grade 5

Standards: Grade 6

Standards: Grade 7

Standards: Grade 8

Standards: Secondary I

Standards: Secondary II

Secondary Mathematics: Pathways

Standards: Secondary III (All Students)

Standards: Secondary III (Calc Pathway)

Standards: Secondary III (Data Science Pathway)

Introduction Utah's Core Mathematics Standards assist in opening doors of opportunity for all students. They set clear expectations and guide students through a focused and coherent progression of math concepts, considering age-level development and emphasizing proficiency of identified essential competencies. These standards establish a strong mathematical foundation in early grades that allows students to engage with complex mathematical ideas as they progress through secondary mathematics.

All students must have the opportunity to learn and meet high mathematics standards in order to understand and navigate the modern world. Engaging with mathematics helps students develop tools to analyze patterns, quantify relationships, and make informed decisions. Understanding mathematics provides students with skills that support success in college, career, and family life.

The standards set grade-specific expectations but do not dictate curriculum or teaching methods, nor do they define the strategies necessary to meet diverse learners' needs (53E-4-202). Customized supports must be provided to ensure all learners have access to the grade-level content in Utah’s Core Mathematics Standards.

Organization of the Standards

In Utah State Code 53E-4-202, the State Board is charged with establishing core standards for Utah Public Schools that identify the basic knowledge, skills, and competencies each student is expected to acquire or master as the student advances through the public education system. These basic knowledge, skills, and competencies increase in depth and complexity from year to year.

Strands

The Utah Core Standards are organized into strands, which represent significant areas of learning within content areas. In mathematics, the strands are organized by overarching themes but do not necessarily represent isolated topics.

Mathematical concepts are connected across multiple strands and grade levels. The https://le.utah.gov/xcode/Title53E/Chapter4/53E-4-S202.html https://le.utah.gov/xcode/Title53E/Chapter4/53E-4-S202.html trajectory of the strands is designed to follow a logical learning progression.

Standards

Within each strand are standards. A standard articulates the knowledge and skills to be obtained by students. A standard represents the essential elements of the expected learning.

Mathematical Skills

In each grade-level core, there is a collection of Mathematical Skills. Each skill is a fundamental mathematical practice that students build as they engage with mathematical content. To ensure students develop capacity with these skills, they have been embedded into every standard across all strands and can be identified by the use of italic text.

Although specific Mathematical Skills have been intentionally identified and connected to each standard, instruction and practice of other Mathematical Skills not referenced in an individual standard may be appropriate if other skills support student learning in mathematics learning experience.

Essential Competencies

Essential competencies have been identified at each grade level to represent foundational understandings that students need to succeed as they continue in future grade levels. The list of competencies is not exhaustive of every standard;

students will engage with all grade-level standards. Educators may prioritize instructional time and, as needed, provide additional instruction or interventions based on the standards that support these competencies to ensure students are proficient.

Shifts in the Standards Changes in the standards were made to improve their consistency, clarity, and progression through the grade levels. These changes include:

● Standards were clarified, and the overall number of standards was reduced.

● Mathematical Skills were identified and are highlighted throughout the standard P-12.

● Expectations were defined for the mathematical skills at each grade level.

● Essential Competencies were identified for each grade level specifying the success criteria associated with proficiently completing the major work of that grade level.

● Data Science standards were emphasized and incorporated into all grades to build statistical literacy essential for Utah citizens.

○ At every grade level, the Data Science standards follow a structure where students model with mathematics by:

■ Asking statistical questions,

■ Collecting and exploring data,

■ Analyzing data,

■ And interpreting results

● In Secondary Math III, specific standards that are essential for all students were highlighted, and additional standards were developed to provide students and their families agency in choosing which math course pathways will best support their post-high school goals.

Building Procedural Fluency These standards define what students should understand and be able to do in their study of mathematics. Throughout the progression of the Utah mathematics core standards, special attention was paid to ensuring that students develop a strong foundation in conceptual understanding and procedural fluency. Procedural fluency is defined as the ability to apply procedures efficiently, flexibly, and accurately, transfer procedures to different problems and contexts, build or modify procedures from other procedures, and recognize when one strategy or procedure is more appropriate to apply than another.

Through the progression of the standards, students are provided opportunities to make sense of and build a conceptual understanding of the mathematical operations. This conceptual understanding supports children in becoming automatic in their retrieval of basic math facts. That basic fact fluency can then be applied as students develop computational and procedural fluency using a variety of numerical strategies and efficient algorithms.

Mathematical Tools and Technology An excellent mathematics program integrates the use of mathematical tools and technology as essential resources to empower students to meet the learning intentions of the classroom. Software and online programs cannot replace authentic interactions between teachers and students, and the value of student-to-student discourse and collaboration. Technology and tools should be used in ways that promote mathematical reasoning and sense-making.

Mathematical Skills The mathematical skills included in Utah’s P-12 Core Mathematics Standards describe mathematical practices that Utah students develop throughout their educational experiences. Students become proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills. Each grade level’s standards have a description in their introductory material specifying the ways students will develop these skills as they learn math concepts. Although specific Mathematical Skills have been intentionally identified and connected to each standard, instruction and practice of other Mathematical Skills not referenced in that standard are encouraged. Below, general descriptions are provided for each skill.

Descriptions of what the skills look like in each grade band are included in the grade level standards. More details about the skills and how they are integrated throughout the standards are included in each grade level’s Utah Mathematics Core Guides.

(Utah Mathematics Core Guides will be updated after these standards are approved by the Utah State Board of Education)

Skill 1: Describe and represent structures, patterns, and relationships

Students learn the language of mathematics through exploration (making sense of a mathematical task), observation (noticing patterns and relationships in mathematics), and highlighting structure (using mathematical tools to organize those relationships). Students use their understanding to make generalizations about the structure of numbers to become more flexible and efficient in applying mathematical ideas.

Students build these skills when they have opportunities to explore new topics collaboratively and independently. As they surface patterns and relationships, they apply mathematical structures to organize ideas and work toward making mathematical generalizations.

Skill 2: Build and use models

Mathematical models use structures to represent relationships. Students build and use verbal, contextual, visual, symbolic, and physical models to represent and make sense of the relationships between quantities in real-world situations and mathematical problems.

Using models enhances student learning by providing students with multiple ways to represent and understand mathematical concepts, connect mathematical ideas to real-world situations, and develop problem-solving skills. Students explore the idea that various models can showcase mathematical thinking in valuable ways.

Using multiple models helps students notice the connections between mathematical structures and promotes the development of mathematical https://www.schools.utah.gov/curr/mathematics/core#Mathematics%20Core%20Guides generalizations that can be leveraged as tools to make sense of and persevere in solving novel problems.

Skill 3: Attend to precision and reasonableness Students exercise mathematical precision and reasonableness in their communication, calculations, and processes. They calculate flexibly, accurately, and efficiently, using clear and concise notation to record their work. They regularly evaluate whether actions, ideas, processes, and solutions make sense and are reasonable.

Precision in communication and calculation fosters a deeper understanding of mathematical concepts. Using precise language, students articulate their thoughts clearly, enhancing their ability to explain their reasoning to others and understand others’ ideas. Accurate and efficient calculations and precise notation enable students to solve problems effectively and minimize errors. The continuous evaluation of reasonableness ensures that students improve their ability to identify and learn from mathematical errors.

Skill 4: Construct, justify, and communicate clear and reasonable arguments

Students construct arguments using the mathematical reasoning that underlies a strategy, solution, or conjecture. They justify their conjectures and clearly communicate their reasoning using concrete referents such as objects, drawings, diagrams, and actions. They justify their arguments with peers through discourse, refining their thinking to create more robust and precise arguments.

Constructing, justifying, and communicating arguments fosters a deep understanding of mathematical concepts. By defending their conjectures, students solidify their mathematical understanding and identify potential flaws in their reasoning. Clear communication enables students to articulate their thoughts effectively, enhancing their ability to explain their reasoning to others. Discourse with peers exposes students to diverse perspectives, promoting critical thinking and collaborative problem-solving.

Skill 5: Add or remove context to make sense of mathematics Students add or remove context to help make sense of mathematical ideas.

Students add context by connecting numbers and operations to images, stories, objects, and events. They remove context from a situation by representing the mathematics using numbers, symbols, and equations. Students visualize complicated concepts as being composed of simpler parts. They identify quantities in a contextual situation, use mathematical models to represent and analyze the relationships between those quantities, and draw conclusions.

When students contextualize and decontextualize mathematical ideas, they can make sense of underlying structures by connecting representations. Working in and out of context throughout a learning progression empowers students to make sense of mathematics and persevere as they explore abstract and real-world applications. This process allows students to take their knowledge of mathematics and meaningfully apply it to their lives.

Skill 6: Ask questions to explore mathematical ideas Students examine mathematical concepts and formulate questions based on what they notice and wonder about given situations. During mathematical discourse, students ask probing questions of one another as they examine arguments and seek to understand others’ ideas.

Asking questions cultivates students’ curiosity about mathematical relationships and helps them refine their understanding. As students build their capacity to ask probing questions to their peers, they engage in more meaningful discourse about mathematics. When students ask and explore rich questions, it advances the mathematical understanding of all learners.

Skill 7: Make conjectures and evaluate the results Students make sense of mathematics when they engage in mathematical investigations. They use their understanding of structures, patterns, and relationships to speculate about the nature of a relationship and then test their conjecture to determine how to proceed. As they test their mathematical hypotheses and evaluate the outcomes of their investigation, they build mathematical generalizations.

Developing and assessing conjectures encourages active engagement and critical thinking. Conjecturing acts as an entry point to exploration for all students. When making conjectures, students have opportunities to continually revise their thinking. Students formulate conjectures, test them, and refine their understanding based on the outcomes. Students view mistakes as assets that help them better refine their mathematical understanding. This process fosters a deeper understanding of underlying mathematical principles. It promotes curiosity, students’ ownership of their learning, and the development of positive mathematical identities.

Skill 8: Select and use tools appropriately and strategically Students strategically select and use relevant and valuable tools for mathematical tasks with which they engage. Tools can be physical objects (e.g., compasses or manipulatives), technological tools (e.g. graphing calculators or virtual manipulatives) or mathematical constructs (e.g., diagrams, strategies, or algorithms).

Students consider the available mathematical problem-solving tools and choose the most relevant and useful ones. When students have the agency to select mathematical tools strategically, it empowers them to visualize, represent, and understand mathematical concepts effectively, fostering critical thinking, problem-solving flexibility, perseverance, and engagement.

Standards: Preschool Age 3

Introduction In Preschool Age 3 (P3) and Preschool Age 4 (P4), the major work of both grades focuses on developing an understanding of the foundational concepts of counting and cardinality and developing the ability to describe and compare measurable attributes of objects and shapes.

In P3, instructional time should focus on three major works: (1) developing concepts of counting and cardinality; (2) beginning to recognize simple patterns;

and (3) beginning to describe shapes and objects. Quality learning environments focus on hands-on experiences during play and interaction with others to incorporate well-designed mathematical experiences aligned to specific learning goals.

Mathematical Skills Link to Detailed Skills Description

Skill 1: Describe and represent structures, patterns, and relationships In grades P-K, students explore patterns and structures in counting and with shapes. Students begin exploring the concepts of addition and subtraction with small numbers.

Skill 2: Build and use models In grades P-2, students use a variety of models to build their understanding of numbers with increasing complexity, leading to an understanding of the base ten place value system. They model the actions and comparisons represented by the operations of addition and subtraction. Consistent exposure to these models supports fluency and helps students connect concrete experiences to foundational mathematical ideas.

Skill 3: Attend to precision and reasonableness https://docs.google.com/document/d/1Sp3cOBG_drOOSms5tf5-TZkJEWv5sNyuziy9MuOx6zs/edit?tab=t.tvyq89rpym1m#heading=h.gj5tmrngm7va

In grades P-K, students use informal language as they develop and learn precise mathematical vocabulary. They begin to learn strategies for assessing their accuracy when counting, naming numbers, and describing shapes.

Skill 4: Construct, justify, and communicate clear and reasonable arguments

In grades P3 and P4, students are learning to communicate their ideas and listen to the ideas of others. They develop strategies to agree and disagree respectfully.

Students begin to explain mathematical ideas.

Skill 5: Add or remove context to make sense of mathematics

In grades P-2, students build an understanding of what the numbers, symbols, pictures, words, etc., in their work represent. Students develop flexibility in moving between concrete, pictorial, and abstract representations of mathematical ideas.

They begin moving back and forth between a problem’s context and its representation to use the form that best fits the situation.

Skill 6: Ask questions to explore mathematical ideas

In grades P-2, students ask questions to help them explore numbers, place value, the properties of addition and subtraction, and the properties of shapes. Students begin to ask questions about data related to the classroom and their day-to-day lives.

Skill 7: Make conjectures and evaluate the results

In grades P-2, students will make and test conjectures related to numbers, place value, the properties of addition and subtraction, and the attributes of shapes. As students explore their conjectures, they will learn to evaluate the reasonableness of their solution path and adjust as needed. Providing students opportunities to revise their thinking helps students develop confidence in problem solving.

Skill 8: Select and use tools appropriately and strategically

In grades P-2, students use tools when they help them make sense of mathematics.

Tools can include, but are not limited to, physical tools, visual tools, and self-created tools such as counters, ten frames, place value charts, hundreds charts, open number lines, strategies, algorithms, etc. Consistent, hands-on practice with these tools builds confidence and prepares students for flexible application.

Standards: Preschool Age 3 (P3)

More details about the standards, including relevant vocabulary, representations that support student learning, pre-requisite skills, and how the skills are integrated, can be found in the Utah Mathematics Core Guides.

(Utah Mathematics Core Guides will be updated after these standards are approved by the Utah State Board of Education)

Counting and Cardinality (CC)

Identify numerals by name, count to 10, use one-to-one correspondence, describe quantities of objects counted, use ordinal language, and subitize.

P3.CC.1

Attend to structure and precise language by reciting number names in sequence to

10. (Sk 1, 3)

P3.CC.2

Describe and represent mathematical structures by beginning to recognize written numbers to represent a quantity. Use a variety of tools to build and model numbers.

(Sk 1, 2, 8)

P3.CC.3

Demonstrate one-to-one correspondence by precisely counting and pointing/touching a line of 1-5 objects. (Sk 3)

P3.CC.4

Begin to use precise number names to identify the number of objects in a group of up to 5 objects, recognizing that the last number in a count represents the total number of objects. (Sk 3)

P3.CC.5

With support, use mathematical language to precisely identify whether the number of objects in one group is the same as those in another group for sets of up to 4 objects. (Sk 3)

P3.CC.6

Identify whether an object is "first" or "second" in a sequence by noticing ordinal structure. Begin to recognize ordering words in everyday contexts. (Sk 5)

P3.CC.7

Use patterns and structures to subitize quantities up to 3. (Sk 1)

Operations and Algebraic Thinking (OA) Identify and explore simple patterns.

P3.OA.1

Describe and model simple patterns in the environment and begin to duplicate and extend those patterns. (Sk 1, 2)

Measurement and Geometry (MG) Identify and describe measurable attributes of basic shapes.

P3.MG.1

With prompting and support, identify and describe measurable attributes (for example, big, small, tall, short). (Sk 1)

P3.MG.2

With prompting and support, make and evaluate conjectures about the measurable attributes of objects and shapes. (Sk 7)

P3.MG.3

With prompting and support, identify, describe, and represent basic shapes and their attributes. (Sk 1)

Data Science (D)

Begin asking questions that lead to collecting and analyzing data by sorting and counting objects.

P3.D.1 With prompting and support, ask questions about people or objects in the classroom. (Sk 6)

P3.D.2 With support, select and use tools to collect and sort data by attributes (color, shape, size). (Sk 8)

P3.D.3

With prompting and support, attend to precision and reasonableness when analyzing data by counting the number of objects in each category (no more than five objects per category). (Sk 3)

Essential Competencies Essential competencies have been identified at each grade level to represent foundational understandings that students need to succeed as they continue in future grade levels. This list is not exhaustive; students will engage with all grade-level standards. Educators may prioritize instructional time and, as needed, provide interventions based on the standards that support these competencies to ensure students are proficient. Through engagement with the knowledge and skills embedded within the standards, by the end of P3, children will be able to:

● Count up to 5 objects in a set, then tell how many objects are in the set without counting again. (CC)

● Count to 10 by ones and recognize that numbers have a known sequence (for example, “1, 2, 3, 4, 5. What comes next?”). (CC)

● Quickly recognize small groups of up to 3 objects without counting. (CC)

● Describe and model simple patterns. (OA)

● Identify and describe basic shapes and their measurable attributes. (MG)

● Begin sorting and counting objects. (D)

Standards: Preschool Age 4

Introduction In Preschool Age 3 (P3) and Preschool Age 4 (P4), the major work of both grades focuses on developing an understanding of foundational concepts of counting and cardinality and developing the ability to describe and compare measurable attributes of objects and shapes.

In P4, instructional time should focus on three major works: (1) reinforcing concepts of counting and cardinality; (2) exploring concepts of addition and subtraction; and

(3) exploring and describing shapes, patterns, and measurable attributes of objects.

Quality learning environments focus on hands-on experiences during play and interaction with others to incorporate well-designed mathematical experiences aligned to specific learning goals.

Mathematical Skills Link to Detailed Skills Description

Skill 1: Describe and represent structures, patterns, and relationships In grades P-K, students explore patterns and structures in counting and with shapes. Students begin exploring the concepts of addition and subtraction with small numbers.

Skill 2: Build and use models

In grades P-2, students use a variety of models to build their understanding of numbers with increasing complexity, leading to an understanding of the base ten place value system. They model the actions and comparisons represented by the operations of addition and subtraction. Consistent exposure to these models supports fluency and helps students connect concrete experiences to foundational mathematical ideas.

https://docs.google.com/document/d/1Sp3cOBG_drOOSms5tf5-TZkJEWv5sNyuziy9MuOx6zs/edit?tab=t.tvyq89rpym1m#heading=h.gj5tmrngm7va

In grades P-K, students use informal language as they develop and learn precise mathematical vocabulary. They begin to learn strategies for assessing their accuracy when counting, naming numbers, and describing shapes.

Skill 4: Construct, justify, and communicate clear and reasonable arguments In grades P3 and P4, students are learning to communicate their ideas and listen to the ideas of others. They develop strategies to agree and disagree respectfully.

Students begin to explain mathematical ideas.

Skill 5: Add or remove context to make sense of mathematics In grades P-2, students build an understanding of what the numbers, symbols, pictures, words, etc., in their work represent. Students develop flexibility in moving between concrete, pictorial, and abstract representations of mathematical ideas.

They begin moving back and forth between a problem’s context and its representation to use the form that best fits the situation.

Skill 6: Ask questions to explore mathematical ideas

In grades P-2, students ask questions to help them explore numbers, place value, the properties of addition and subtraction, and the properties of shapes. Students begin to ask questions about data related to the classroom and their day-to-day lives.

Skill 7: Make conjectures and evaluate the results In grades P-2, students will make and test conjectures related to numbers, place value, the properties of addition and subtraction, and the attributes of shapes. As students explore their conjectures, they will learn to evaluate the reasonableness of their solution path and adjust as needed. Providing students opportunities to revise their thinking helps students develop confidence in problem solving.

Skill 8: Select and use tools appropriately and strategically

In grades P-2, students use tools when they help them make sense of mathematics.

Tools can include, but are not limited to, physical tools, visual tools, and self-created tools such as counters, ten frames, place value charts, hundreds charts, open number lines, strategies, algorithms, etc. Consistent, hands-on practice with these tools builds confidence and prepares students for flexible application.

Standards: Preschool Age 4 (P4)

More details about the standards, including relevant vocabulary, representations that support student learning, pre-requisite skills, and how the skills are integrated, can be found in the Utah Mathematics Core Guides.

(Utah Mathematics Core Guides will be updated after these standards are approved by the Utah State Board of Education)

Counting and Cardinality (CC)

Count to 20, recognize numerals, demonstrate one-to-one correspondence, compare quantities, use ordinal language, and subitize.

P4.CC.1

With support, attend to mathematical structure and precisely recite number names and the counting sequence to 20. (Sk 1, 3)

P4.CC.2

Describe and represent structures and patterns by associating a quantity with written numerals up to 10. Begin to use a variety of tools to form numerals and represent quantities. (Sk 1, 8)

P4.CC.3

Demonstrate one-to-one correspondence by precisely counting and pointing to or touching one object per count for an arrangement of up to 10 objects. (Sk 3)

P4.CC.4

Use precise number names to identify the number of objects in a group of up to 10, recognizing that the last number in a count represents the total number of objects.

(Sk 3)

P4.CC.5

Use mathematical language to precisely identify whether the number of objects in one group is more than, less than, or the same as the number of objects in another group for sets of up to 6 objects. (Sk 3)

P4.CC.6

Use ordering words to precisely identify and state whether an object is "first,” "second," "third," or "last" in a sequence. Begin to apply ordering words to everyday contexts. (Sk 3, 5)

P4.CC.7

Use patterns and structures to subitize quantities of up to five. (Sk 1)

Operations and Algebraic Thinking (OA)

Identify and manipulate simple patterns and represent addition and subtraction problems involving numbers up to and including 5.

P4.OA.1

Build and use models to explore and represent real-world situations involving addition (adding to or putting together) and subtraction (taking from or taking apart) with numbers up to and including 5. (Sk 2)

P4.OA.2

Select and use tools to decompose numbers less than or equal to five. Make a conjecture about why a number or set of objects can be broken apart in multiple ways. (Sk 7, 8)

P4.OA.3

Select and use tools to make sums of 5 using quantities from 0–5. (Sk 8)

P4.OA.4

Describe and represent simple patterns by duplicating, extending, and creating patterns. (Sk 1)

Measurement and Geometry (MG)

Identify, describe, build, and compare basic shapes and their measurable attributes.

P4.MG.1

Describe objects using vocabulary specific to measurable attributes (for example, length [long/short], weight [heavy/light], size [big/small], and distance [near/far]).

(Sk 1)

P4.MG.2

Test and make conjectures to directly compare two objects using measurable attributes. (Sk 7)

P4.MG.3

Identify, describe, and represent two- and three-dimensional shapes regardless of size and orientation. Describe objects in the environment by using names of shapes, and identify the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. (Sk 1)

P4.MG.4

Build and use models to create basic shapes using a variety of tools. Explore combining basic shapes to create new shapes. (Sk 2)

Data Science (D)

Ask questions that lead to collecting, analyzing, and interpreting data by sorting and counting objects.

P4.D.1 With prompting and support, ask questions about a given topic to investigate situations leading to collecting and analyzing data within the classroom. (Sk 6)

P4.D.2 Select and use tools to collect and sort data by explicit attributes (color, shape, size).

(Sk 8)

P4.D.3 With prompting and support, attend to precision and reasonableness when analyzing data by counting the number of objects in each category. (Sk 3)

P4.D.4 With prompting and support, construct, justify, and communicate clear and reasonable arguments when interpreting data to identify which groups contain more, less, or the same amount. (Sk 4)

Essential Competencies Essential competencies have been identified at each grade level to represent foundational understandings that students need to succeed as they continue in future grade levels. This list is not exhaustive; students will engage with all grade-level standards. Educators may prioritize instructional time and, as needed, provide interventions based on the standards that support these competencies to ensure students are proficient. Through engagement with the skills and knowledge embedded within the Utah Core Mathematics Standards, by the end of P4, children will be able to:

● Count up to 10 objects in a set, then tell how many objects are in the set without counting again. (CC)

● Begin to connect number names and quantities with written numerals (CC)

● Count to 20. (CC)

● Quickly recognize groups of up to 5 objects without counting. (CC)

● Create and extend simple patterns. (OA)

● Understand and model addition and subtraction with numbers up to and including 5 with concrete objects, fingers, movement, and simple drawings.

(OA)

● Identify, describe, build, and compare basic shapes using their measurable attributes. (MG)

● Use positional language to describe objects in the environment (above, below, next to, etc.) (MG)

● Begin interpreting data by sorting and counting objects (D)

Standards: Kindergarten (K)

Introduction

In kindergarten through second grade, the major work of each grade focuses on developing an understanding of addition and subtraction and building an understanding of the base-ten place value number system. Students begin to develop an understanding of foundational concepts of linear measurement and explore the defining attributes of shapes.

In kindergarten, instructional time should focus on three major works: (1) mastering concepts of counting and cardinality; (2) representing, relating, and operating on whole numbers, initially with sets of objects; and (3) describing shapes and space.

More learning time in kindergarten should be devoted to number sense than to other topics, as students build fluency through concrete experiences, visual models, and verbal expressions .

Mathematical Skills Link to Detailed Skills Description

Skill 1: Describe and represent structures, patterns, and relationships

In grades P-K, students explore patterns and structures in counting and with shapes. Students begin exploring the concepts of addition and subtraction with small numbers.

Skill 2: Build and use models

In grades P-2, students use a variety of models to build their understanding of numbers with increasing complexity, leading to an understanding of the base ten place value system. They model the actions and comparisons represented by the operations of addition and subtraction. Consistent exposure to these models supports fluency and helps students connect concrete experiences to foundational mathematical ideas.

https://docs.google.com/document/d/1Sp3cOBG_drOOSms5tf5-TZkJEWv5sNyuziy9MuOx6zs/edit?tab=t.tvyq89rpym1m#heading=h.gj5tmrngm7va

In grades P-K, students use informal language as they develop and learn precise mathematical vocabulary. They begin to learn strategies for assessing their accuracy when counting, naming numbers, and describing shapes.

Skill 4: Construct, justify, and communicate clear and reasonable arguments

In grades K-2, students are learning to communicate their ideas and listen to the ideas of others. They develop strategies to agree and disagree respectfully. They begin to justify their arguments with evidence and draw connections between mathematical representations. Students also begin to compare and contrast their strategies and solutions to others’.

Skill 5: Add or remove context to make sense of mathematics

In grades P-2, students build an understanding of what the numbers, symbols, pictures, words, etc., in their work represent. Students develop flexibility in moving between concrete, pictorial, and abstract representations of mathematical ideas.

They begin moving back and forth between a problem’s context and its representation to use the form that best fits the situation.

Skill 6: Ask questions to explore mathematical ideas

In grades P-2, students ask questions to help them explore numbers, place value, the properties of addition and subtraction, and the properties of shapes. Students begin to ask questions about data related to the classroom and their day-to-day lives.

Skill 7: Make conjectures and evaluate the results

In grades P-2, students will make and test conjectures related to numbers, place value, the properties of addition and subtraction, and the attributes of shapes. As students explore their conjectures, they will learn to evaluate the reasonableness of their solution path and adjust as needed. Providing students opportunities to revise their thinking helps students develop confidence in problem solving.

Skill 8: Select and use tools appropriately and strategically

In grades P-2, students use tools when they help them make sense of mathematics.

Tools can include, but are not limited to physical tools, visual tools, and self-created tools such as counters, ten frames, place value charts, hundreds charts, open number lines, strategies, algorithms, etc. Consistent, hands-on practice with these tools builds confidence and prepares students for flexible application.

Standards: Kindergarten (K)

More details about the standards, including relevant vocabulary, representations that support student learning, pre-requisite skills, and how the skills are integrated, can be found in the Utah Mathematics Core Guides.

(Utah Mathematics Core Guides will be updated after these standards are approved by the Utah State Board of Education)

Counting and Cardinality (CC)

Know number names and the counting sequence. Count to tell the number of objects and represent a given quantity in a set. Identify and compare quantities of objects and numerals. Identify an object’s ordinal position. Subitize quantities.

K.CC.1

Attend to structure and patterns by precisely reciting numbers in sequence from 1 to 100 by ones and tens. (Sk 1, 3)

K.CC.2

Attend to structure and patterns when counting forward from a given number between 1 and 100. (Sk 1)

K.CC.3

Use precise language and actions to identify and justify the quantity of objects in a group of 0–20. (Sk 3, 4)

K.CC.4

Precisely read and write numbers 0 to 20 using base-ten numerals. Represent a quantity of 0 to 20 objects with written numerals. (Sk 3)

K.CC.5

Identify and compare quantities of objects and numerals up to 10 using the words “greater than,” “less than,” and “equal to.” Use reasoning to justify comparisons. (Sk 4)

K.CC.6

Use precise mathematical language to identify and justify an object's ordinal position.

Order quantities or numerals using the ordinal words “first” through “tenth.” (Sk 3, 4)

K.CC.7

Use patterns and structures to subitize quantities up to 10. (Sk 1)

Operations and Algebraic Thinking (OA)

Understand addition as putting together and adding to, and subtraction as taking apart and taking from.

K.OA.1

Select and use tools to represent addition and subtraction with sums and minuends up to and including 10. Attend to the precision and reasonableness of results. (Sk 3, 8)

K.OA.2

Build and use models to solve addition and subtraction story problems with sums or minuends up to and including 10. Add or remove story context to solve problem type situations of adding to and taking from with result unknown, putting together with the total unknown, and putting together with both addends unknown. (Sk 2, 5)

K.OA.3

Build and use models to decompose numbers less than or equal to 10 into pairs in multiple ways. Make conjectures about why a number or set of objects can be broken apart in multiple ways. (Sk 2, 7)

K.OA.4

Attend to precision and reasonableness when using strategies to add and subtract flexibly, accurately, and efficiently using numbers less than or equal to 5. (Sk 3)

Number and Operations in Base Ten (NBT) Build and use models to compose and decompose numbers from 11-19 into ten ones and some more ones to conceptually understand place value.

K.NBT.1

Build and use models to compose and decompose numbers from 11-19 into ten ones and some more ones. Represent structure from those models with math language and an equation (e.g., "Eighteen is the same as 10 and 8: 18=10+8).

Describe the pattern that teen numbers are always composed of ten ones and some more ones. (Sk 1, 2)

Measurement and Geometry (MG) Identify, describe, and compare measurable attributes of objects. Analyze, compare, create, and compose shapes.

K.MG.1

Attend to precision and reasonableness when describing several measurable attributes of a single object. Make conjectures to compare two objects with a measurable attribute to see which object has "more of”/"less of" the attribute and evaluate the results of those conjectures. (Sk 3, 7)

K.MG.2

Describe structures, names, and attributes of two-dimensional and three-dimensional shapes regardless of size or orientation, and describe their relative positions in the environment. (Sk 1)

K.MG.3

Make conjectures and evaluate the results to analyze, compare, and sort two- and three-dimensional shapes and objects in different sizes and orientations, using informal language to describe their similarities, differences, and other attributes (for example, color, size, shape, number of sides). (Sk 7)

K.MG.4

Build and use models to create two- and three-dimensional shapes and objects. Use simple shapes to form a variety of shapes. (Sk 2)

Data Science (D)

Ask questions that lead to collecting, analyzing, and interpreting data. Sort, count, and compare objects in a data set.

K.D.1

Ask questions about a given topic to investigate situations that will lead to collecting and analyzing data within the classroom. (Sk 6)

K.D.2

Select and use tools appropriately and strategically to collect and organize data with up to three categories. (Sk 8)

K.D.3

Attend to precision and reasonableness when analyzing data by counting the number of objects in each category. (Sk 3)

K.D.4

Construct, justify, and communicate clear and reasonable arguments when interpreting data to identify which groups contain more, less, or the same amount using category quantities. (Sk 4)

Essential Competencies Essential competencies have been identified at each grade level to represent foundational understandings that students need to succeed as they continue in future grade levels. This list is not exhaustive; students will engage with all grade-level standards. Educators may prioritize instructional time and, as needed, provide interventions based on the standards that support these competencies to ensure students are proficient. Through engagement with the skills and knowledge embedded within the Utah Core Mathematics Standards, by the end of kindergarten, children will be able to:

● Count up to 20 objects in a set to answer questions like "how many." (CC)

● Read and write numbers 0-20. (CC)

● Count to 100 by ones and tens. (CC)

● Compare quantities of objects and numerals up to 10. (CC)

● Quickly recognize groups of up to 10 objects without counting. (CC)

● Fluently add and subtract when the largest number in the equation is no greater than 5. (Basic fact fluency) (OA)

● Draw pictures or use objects to solve addition and subtraction story problems where the largest number in the story is no greater than 10. (OA)

● Break apart numbers less than or equal to 10 in multiple ways. (OA)

● Break apart numbers 11-19 into ten ones and some more ones. (NBT)

● Compare and sort objects and shapes using attributes and measurable characteristics. (MG)

● Interpret data by sorting, counting, and comparing objects in a data set. (D)

Standards: Grade 1

Introduction In kindergarten through second grade, the major work of each grade focuses on developing an understanding of addition and subtraction and building an understanding of the base-ten place value number system. Students develop an understanding of foundational concepts of linear measurement and explore the defining attributes of shapes.

In Grade 1, instructional time should focus on four major works: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction;

(2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes.

Mathematical Skills

Link to Detailed Skills Description

Skill 1: Describe and represent structures, patterns, and relationships

In grades 1-2, students explore patterns and structures such as place value, properties of operations, and composition of shapes. They use structures and patterns to develop a deep understanding of addition and subtraction. Students use structures and patterns to see complicated things as single objects or as composed of several objects.

Skill 2: Build and use models

In grades P-2, students use a variety of models to build their understanding of numbers with increasing complexity, leading to an understanding of the base ten place value system. They model the actions and comparisons represented by the operations of addition and subtraction. Consistent exposure to these models supports fluency and helps students connect concrete experiences to foundational https://docs.google.com/document/d/1Sp3cOBG_drOOSms5tf5-TZkJEWv5sNyuziy9MuOx6zs/edit?tab=t.tvyq89rpym1m#heading=h.gj5tmrngm7va

Skill 3: Attend to precision and reasonableness

In grades 1-2, students refine informal language as they develop and learn precise mathematical vocabulary. Students use structure to accurately record, label, and share their thinking. They explore when a precise answer is necessary and when making an estimate, which can help them determine if a solution makes sense.

Students use estimation strategies based on place value structure and the properties of operations to assess reasonableness throughout the problem-solving process.

Skill 4: Construct, justify, and communicate clear and reasonable arguments

In grades K-2, students are learning to communicate their ideas and listen to the ideas of others. They develop strategies to agree and disagree respectfully. They begin to justify their arguments with evidence and draw connections between mathematical representations. Students also begin to compare and contrast their strategies and solutions to others’.

Skill 5: Add or remove context to make sense of mathematics

In grades P-2, students build an understanding of what the numbers, symbols, pictures, words, etc., in their work represent. Students develop flexibility in moving between concrete, pictorial, and abstract representations of mathematical ideas.

They begin moving back and forth between a problem’s context and its representation to use the form that best fits the situation.

Skill 6: Ask questions to explore mathematical ideas

In grades P-2, students ask questions to help them explore numbers, place value, the properties of addition and subtraction, and the properties of shapes. Students begin to ask questions about data related to the classroom and their day-to-day lives.

Skill 7: Make conjectures and evaluate the results

In grades P-2, students will make and test conjectures related to numbers, place value, the properties of addition and subtraction, and the attributes of shapes. As students explore their conjectures, they will learn to evaluate the reasonableness of their solution path and adjust as needed. Providing students opportunities to revise their thinking helps students develop confidence in problem solving.

Skill 8: Select and use tools appropriately and strategically

In grades P-2, students use tools when they help them make sense of mathematics.

Tools can include, but are not limited to physical tools, visual tools, and self-created tools such as counters, ten frames, place value charts, hundreds charts, open number lines, strategies, algorithms, etc. Consistent, hands-on practice with these tools builds confidence and prepares students for flexible application.

Standards: Grade 1

More details about the standards, including relevant vocabulary, representations that support student learning, pre-requisite skills, and how the skills are integrated, can be found in the Utah Mathematics Core Guides.

(Utah Mathematics Core Guides will be updated after these standards are approved by the Utah State Board of Education)

Operations and Algebraic Thinking (OA)

Represent and solve problems involving addition and subtraction with sums and minuends less than or equal to 20. Understand and apply properties of operations and the relationship between addition and subtraction. Understand the meaning of the equal sign.

1.OA.1

Build and use models to solve addition and subtraction story problems with sums and minuends less than or equal to 20. Add or remove story context to solve problem type situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions except start unknown. (Sk 2, 5)

1.OA.2

Build and use models to solve story problems involving adding three whole numbers with sums less than or equal to 20. Add or remove context to make sense of problems and solutions. (Sk 2, 5)

1.OA.3

Make and test conjectures about properties of addition. Add context and compare arguments to make sense of these properties as strategies to add and subtract. (Sk 4, 5, 7)

1.OA.4

Describe and represent subtraction as an unknown-addend problem. Construct and justify arguments that explain how to determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. (Sk 1, 4)

1.OA.5

Describe structures and relationships to connect counting to addition and subtraction. (Sk 1)

1.OA.6

Justify strategies used to add and subtract with sums and minuends less than or equal to 20, and compare strategies with others. (Sk 4)

1.OA.7

Attend to precision and reasonableness when using strategies to add and subtract flexibly, accurately, and efficiently with sums and minuends less than or equal to

10. (Sk 3)

1.OA.8

Build and use models to make sense of the meaning of the equal sign. Make conjectures regarding whether addition and subtraction equations are true or false and evaluate the reasonableness of results. (Sk 2, 7)

Number and Operations in Base Ten (NBT) Count and represent numbers up to and including 120. Build and use models to conceptually understand place value. Use place value understanding to add and subtract larger numbers.

1. NBT.1

Count to 120 with precision, starting at any number, and represent a number of objects up to and including 120 with a written numeral. (Sk 3)

1. NBT.2

Use place value understanding to count by tens and describe the pattern in the ones place and tens place when counting by ten. (Sk 1)

1.NBT.3

Build and use models to represent and identify the number of tens and ones in a two-digit number. (Sk 2)

1.NBT.4

Use tools and place value understanding to compare two two-digit numbers.

Represent the relationship between the two numbers using comparison symbols <, =, and >. (Sk 1, 8)

1.NBT.5

Build and use concrete and pictorial models to determine the sum of two two-digit numbers, a two-digit number and a one-digit number, and a two-digit number and a multiple of ten (less than or equal to 90). Use the models to represent and describe the structure of combining tens with tens and ones with ones. Make conjectures about when to compose ten ones into a ten. Ask questions to compare concrete and pictorial models. (Sk 1, 2, 6, 7)

1.NBT.6

Build and use concrete and pictorial models to subtract a multiple of ten from a larger multiple of ten.