Colorado State Standards for Mathematics:

CO.1. Number Sense, Properties, and Operations

1.1. Whole numbers can be used to name, count, represent, and order quantity. Students can:

1.1.a. Count and represent objects to 20

1.1.b. Identify, read, and write corresponding numerals

1.1.c. Compare sets up to 10 objects and use language to describe more, less, or same

1.1.d. Compare two sets of objects to at least 25 using language such as ''more,'' ''less,'' or ''the same''

1.1.e. Identify small groups of objects -fewer than five without counting, including zero as ''no objects''

1.1.f. Estimate quantities less than 20

1.2. Adding and subtracting to 10 involves composing and decomposing using a variety of strategies and representations. Students can:

1.2.a. Use objects including coins, and drawings to model addition and subtraction problems to 10 (PFL)

1.2.b. Identify numbers one more or one less than a given number up to 10

1.2.c. Determine if more than or less than is needed to change one quantity to another

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Patterns can repeat. Students can:

2.1.a. Duplicate a simple pattern

2.1.b. Extend a repeating two-element pattern using a variety of materials such as numbers, letters, shapes, and manipulatives

2.2. Relationships exist between numbers. Students can:

2.2.a. Generalize the counting sequence pattern from counting all to knowing ''one more'' and ''one less''

2.2.b. Communicate the relationship between composing and decomposing numbers

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays of information can used to answer questions. Students can:

3.1.a. Collect classroom data

3.1.b. Identify and compare own data to group's data

3.1.c. Describe bar graphs to answer questions such as more or less and simple trends

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Shapes are described by their characteristics and position. Students can:

4.1.a. Recognize and informally describe two dimensional shapes with varying orientation, sizes, and shapes

4.1.b. Use relational vocabulary, such as above, below and next to, to describe spatial relationships

4.2. Measurement is used to compare and order objects. Students can:

4.2.a. Recognize and compare attributes of length, height, weight, capacity of objects

4.2.b. Use estimates of measurements from everyday experiences

4.2.c. Order several objects by length, height, weight, capacity, or price (PFL)

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The whole number system describes place value relationships from ones to 100 and forms the foundation for efficient algorithms. Students can:

1.1.a. Count, read, and write numbers to 100

1.1.b. Estimate quantities less than 100

1.1.c. Represent quantities using tens units and ones units

1.1.d. Locate numbers up to 100 on a number display

1.1.e. Compare two sets of objects, including pennies, up to at least 25 using language such as ''three more or three fewer'' (PFL)

1.2. Adding and subtracting involve composing and decomposing using a variety of strategies. Students can:

1.2.a. Use addition when putting sets together and subtraction for breaking sets apart or describing the difference between sets

1.2.b. Use number relationships such as doubles, one more or one less, and the relationship between composing and decomposing to solve addition and subtraction problems

1.2.c. Identify coins and find the value of a collection of two coins (PFL)

1.2.d. Demonstrate fluency with basic addition and related subtraction facts through sums to 10

1.3. Parts of objects can be shown as fractions. Students can:

1.3.a. Identify unit fractions 1/2, 1/3, and 1/4 as parts of wholes or parts of groups

1.3.b. Understand fractions as equal shares or parts

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Patterns can grow. Students can:

2.1.a. Count objects by groups of 2 or 5

2.1.b. Extend a repeating pattern based on a rule

2.2. Number relationships can be used to solve problems. Students can:

2.2.a. Use number relationships such as doubles, or plus or minus one to solve problems

2.2.b. Use the inverse relationship between adding and subtracting to solve problems

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays of data can be created using individual student data. Students can:

3.1.a. Contribute individual data to classroom data display

3.1.b. Read information from picture graphs, bar graphs, and tally charts

3.1.c. Describe data by applying the concepts of largest, smallest and most often

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Shapes can be created and described by composing and decomposing. Students can:

4.1.a. Recognize, describe, and make shapes according to given relationships, attributes, or properties

4.1.b. Sort geometric figures and describe how they are alike and different

4.1.c. Combine and take apart shapes to create new shapes and describe results

4.2. Measurement is used to compare and order objects and events. Students can:

4.2.a. Measure the length of common objects using nonstandard units such as created units, popsicle sticks, or paper clips

4.2.b. Compare and order objects by length and weight

4.2.c. Distinguish units of time (day, night, morning, afternoon, hours) and connect them to common events

4.2.d. Compare and order units of time

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The whole number system describes place value relationships from ones to 1,000 and forms the foundation for efficient algorithms. Students can:

1.1.a. Read and write numbers to 1,000 and identify place value for three-digit numbers

1.1.b. Describe relationships between ones, tens, and hundreds

1.1.c. Explain the value of a digit in a three-digit number

1.1.d. Order a collection of whole numbers

1.2. Formulate, represent, and use algorithms to add and subtract two-digit whole numbers with flexibility, accuracy, and efficiency. Students can:

1.2.a. Demonstrate fluency with basic addition and subtraction facts to sums of 20

1.2.b. Find the value of a collection of coins and choose coins to have a given value

1.2.c. Create stories and models, including linear and difference, to illustrate addition and subtraction

1.2.d. Select and use appropriate methods to estimate sums and differences or calculate them mentally depending on the context and numbers involved

1.2.e. Apply addition and subtraction concepts to financial decision-making (PFL)

1.3. Fractions represent parts of a whole object or set. Students can:

1.3.a. Partition basic shapes, using common fractions such as 1/2, 1/3, and 1/4

1.3.b. Partition sets using common fractions such as 1/2, 1/3, 1/4

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Patterns are based on rules. Students can:

2.1.a. Count objects by groups of 2, 5, and 10

2.1.b. Identify a missing number in a sequence, and describe a rule

2.1.c. Create and extend repeating patterns of 3-5 elements using a variety of materials such as numbers, letters, shapes, and manipulatives

2.2. Number relationships can be used to develop computation strategies. Students can:

2.2.a. Use ten-based strategies to solve addition and subtraction facts to 20

2.2.b. Demonstrate the structure of numbers as tens and ones in addition and subtraction

2.2.c. Communicate the inverse relationship between addition and subtraction, and use this relationship to efficiently solve and check problems

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays of data can be constructed in a variety of formats. Students can:

3.1.a. Construct picture graphs and bar graphs from a data set

3.1.b. Read and explain information in picture graphs and bar graphs

3.1.c. Describe data using concepts of median and range

3.2. Mathematical models are used to describe the likelihood of an outcome or event. Students can:

3.2.a. Collect data using chance devices, such as spinners and describe outcomes as likely or unlikely

3.2.b. Apply the concepts of likely or not likely to decisions from daily life (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Shapes can be created and described by quantifiable attributes. Students can:

4.1.a. Recognize, describe, and create geometric figures according to given quantifiable attributes such as number of sides and size

4.1.b. Identify symmetry in two-dimensional figures

4.1.c. Use quantifiable attributes to describe and estimate size of objects

4.2. Some attributes of objects are measurable and can be quantified using different tools. Students can:

4.2.a. Identify the measurable attribute and appropriate unit of measure for an object

4.2.b. Use common objects as non-standard units

4.2.c. Use standard linear measuring tools to measure to the nearest whole unit

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The whole number system describes place value relationships from ones to 10,000 and forms the foundation for efficient algorithms. Students can:

1.1.a. Read and write numbers from one to 10,000 and explain place value for four-digit numbers

1.1.b. Generalize the change represented when moving from one place to another place in a number

1.1.c. Compose and decompose multi-digit numbers based on place value

1.2. Parts of a whole can be modeled and represented in different ways. Students can:

1.2.a. Use drawings, models, and numerals to represent fractions (halves, thirds, fourths, sixths, eighths) based on a whole shape, number set, or number line

1.2.b. Estimate and justify the reasonableness of solutions to problems involving representations of fractions

1.2.c. Describe why equivalent fractions are two ways of modeling the same quantity using a model or drawing

1.3. Formulate, represent, and use algorithms to add and subtract multi-digit whole numbers with flexibility, accuracy, and efficiency. Students can:

1.3.a. Use number sense to estimate and justify the reasonableness of solutions to problems

1.3.b. Use flexible methods of computing, including student-generated strategies and standard algorithms

1.3.c. Estimate using strategies such as front-end estimation or landmark numbers

1.4. Multiplying and dividing are inverse operations modeled in a variety of ways. Students can:

1.4.a. Demonstrate fluency with multiplication and division facts with single-digit factors

1.4.b. Describe relationships between related facts and between multiplication and division

1.4.c. Represent multiplication and division problems with drawings, models, number sentences, and stories

1.4.d. Model strategies to achieve a personal financial goal using arithmetic operations (PFL)

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Number patterns are based on operations and relationships. Students can:

2.1.a. Extend simple arithmetic and geometric sequences

2.1.b. Count by and analyze patterns in multiples of 2, 3, 5, 9, 10, 11, 25, 50 and 100

2.1.c. Use known multiplication facts to solve unknown multiplication problems

2.2. Number properties can be used to solve problems. Students can:

2.2.a. Use the commutative property to solve addition and multiplication problems

2.2.b. Use the associative property to solve addition problems

2.2.c. Use the relationship between addition and multiplication to solve problems

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays of data can be used to answer questions of interest. Students can:

3.1.a. Compose questions to generate data

3.1.b. Collect and organize data from simple experiments or surveys in class

3.1.c. Create picture graphs, bar graphs, dot plots, and frequency tables from a data set

3.1.d. Describe data using the concepts of mode, clusters and gaps

3.2. Mathematical models are used to explore and describe fairness. Students can:

3.2.a. Investigate chance devices such as coins, spinners, and number cubes

3.2.b. Apply the concepts of impossible, unlikely and likely

3.2.c. Determine if a chance device is fair or unfair

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Geometric figures are described by their attributes and position in the plane. Students can:

4.1.a. Construct and describe two-dimensional shapes by attributes and properties such as sides, angles, and symmetry

4.1.b. Recognize and demonstrate transformations - reflections, translations, and rotations - of basic shapes or designs

4.1.c. Use geometric properties of points and line segments to describe figures

4.2. Objects have distinct attributes that can be measured with appropriate tools. Students can:

4.2.a. Use standard units to measure to the nearest 1/2 or whole inch or centimeter

4.2.b. Estimate and measure distance and perimeter

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The decimal number system describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms. Students can:

1.1.a. Read and write numbers from one to 100,000 and explain place value for five-digit numbers

1.1.b. Compose and decompose multi-digit numbers based on place value

1.1.c. Read and write numbers to the hundredths place

1.1.d. Identify the value of any given digit in a number with decimals to the hundredths place

1.2. Formulate, represent, and use algorithms to multiply and divide with flexibility, accuracy, and efficiency. Students can:

1.2.a. Use flexible and efficient methods of computing including standard algorithms to solve three- or four-digit by one-digit multiplication or division problems

1.2.b. Estimate using strategies such as front end or rounding to justify the reasonableness of solutions to problems

1.2.c. Demonstrate fluency with multiplication facts and their related division facts 0 to 12

1.2.d. Explain why multi-digit multiplication and division procedures work based on place value properties and use them to solve problems

1.3. Different models and representations can be used to compare fractional parts. Students can:

1.3.a. Solve comparison problems using models of fractions with like and unlike denominators through 10

1.3.b. Estimate and justify the reasonableness of solutions to problems involving comparison of fractions

1.3.c. Demonstrate equivalent fractions, decimals, and percents using drawings and models

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Number patterns and relationships can be represented by symbols. Students can:

2.1.a. Use number relationships to find the missing number in a sequence

2.1.b. Use a symbol to represent and find an unknown quantity in a problem situation

2.1.c. Complete input/output tables

2.1.d. Find the unknown in simple equations

2.2. Number properties and relationships can be used to solve problems. Students can:

2.2.a. Use and describe number patterns for counting by 2, 5, 9, 10, and 11 from a given starting number

2.2.b. Communicate the inverse relationship between multiplication and division, and use this relationship to efficiently solve and check problems

2.2.c. Use the commutative and associative properties of multiplication to solve problems

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays of classroom data can be used to summarize information across the content areas. Students can:

3.1.a. Compose questions to generate data related to grade level areas of study

3.1.b. Collect data from class experiments or multi-classroom surveys

3.1.c. Create data displays appropriate to data collected

3.1.d. Describe data using the concept of shape of the distribution

3.2. Mathematical models are used to test predictions about the likelihood of events. Students can:

3.2.a. Formulate a question to test a prediction, and conduct an experiment using chance devices, such as coins, spinners, and number cubes, to test predictions

3.2.b. Represent the outcomes of experiments with fractions, and describe using the concepts of impossible, unlikely, likely, and certain

3.2.c. Describe the likelihood of real-life situations using the concepts of impossible, unlikely, likely and certain (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Geometric figures are described by their attributes and specific location in the plane. Students can:

4.1.a. Identify parallel, perpendicular, and intersecting line segments in the plane and within geometric shapes

4.1.b. Create geometric designs using transformations: reflections, translations, and rotations

4.1.c. Compare geometric figures according to the attributes of congruence, symmetry, and angle size

4.1.d. Name and locate points specified by ordered number pairs on a coordinate grid

4.2. Appropriate measurement tools, units, and systems are used to measure different attributes of objects and time. Students can:

4.2.a. Model area using square units

4.2.b. Distinguish between area and perimeter

4.2.c. Convert using unit equivalencies within the standard measurement system (yards to feet and feet to inches, pounds to ounces, gallons to quarts)

4.2.d. Convert using unit equivalencies within the metric measuring system (meters to centimeters, kilometers to meters, and liters to milliliters)

4.2.e. Estimate and measure elapsed time to the nearest quarter hour

4.2.f. Select an appropriate tool and unit for measuring length, weight, and capacity

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The characteristics of numbers can be used to classify them in various ways. Students can:

1.1.a. Apply concepts of squares, primes, composites, factors, and multiples to solve problems

1.1.b. Use the identity, associative, commutative, and distributive properties to solve problems

1.1.c. Describe and use divisibility rules for two, three, four, five, six, nine, and 10 to solve problems

1.2. In the real number system, commonly used rational numbers have multiple equivalent representations. Students can:

1.2.a. Find equivalent forms of commonly used fractions, decimals, and percents using models, drawings, and computational strategies

1.2.b. Use common fractions and percents to calculate parts of whole numbers in problem situations including comparisons of savings rates at different financial institutions (PFL)

1.2.c. Model addition, subtraction, and multiplication of fractions, decimals, and percents

1.2.d. Compose and decompose multi-digit whole numbers and decimals based on place value

1.2.e. Represent numbers to 1,000,000 with expanded notation and exponents

1.3. Formulate, represent, and use algorithms to multiply and divide multi-digit whole numbers with flexibility, accuracy, and efficiency. Students can:

1.3.a. Use flexible methods of computing including standard algorithms to multiply and divide multi-digit numbers by two-digit factors or divisors

1.3.b. Model multiplication and division using area, linear, and grouping models

1.3.c. Interpret remainders and select the most useful form of the quotient in division problems

1.3.d. Select and use appropriate methods to estimate products and quotients or calculate them mentally depending on the context and numbers involved

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Number patterns and relationships can be described using a variety of tools. Students can:

2.1.a. Analyze and describe patterns and relationships using words, tables, graphs, symbols, and technology

2.1.b. Explain, extend, and use patterns and relationships in solving problems, including those involving saving and checking accounts such as understanding that spending more means saving less (PFL)

2.2. When a relationship exists between two quantities, a change in one results in a change in the other. Students can:

2.2.a. Express change relationships involving whole numbers with if/then statements, input/output boxes, function tables, and rule statements

2.2.b. Select, describe, and use symbols to express unknown quantities

2.2.c. Use patterns to solve problems including those involving saving and checking accounts such as the pattern created when saving $10 a month (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays and summary statistics are used to describe and interpret data. Students can:

3.1.a. Formulate a question and hypothesis to design appropriate data collection and display methods

3.1.b. Select and create appropriate displays of data including double bar graphs, time plots, and line graphs

3.1.c. Interpret data using the concepts of shape of distribution, range, mode, median and mean

3.1.d. Draw conclusions, and make convincing arguments based on categorical and numerical data analysis

3.2. Mathematical models are used to determine probability, analyze and describe the outcomes of events. Students can:

3.2.a. Organize all possible outcomes of events in a list or chart

3.2.b. Use fractions, decimals, and percents to quantify the likelihood of events

3.2.c. Explain why a game involving chance devices such as number cubes or spinners is fair or unfair

3.2.d. Compare individual data to class data collected from chance devices to describe the differences in outcomes based on sample size

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Geometric figures in the plane and in space are described and analyzed by their attributes. Students can:

4.1.a. Relate two-dimensional shapes to three-dimensional shapes using faces, edges, and vertices

4.1.b. Predict and describe the results of transformations: translations, reflections, rotations

4.1.c. Classify and compare angles

4.1.d. Apply concepts of parallel, perpendicular, congruence and line symmetry

4.2. Linear measure, area, and volume are fundamentally different and require different units of measure. Students can:

4.2.a. Accurately measure length to the nearest 1/8 inch or millimeter

4.2.b. Determine the perimeter of polygons and area of rectangles

4.2.c. Distinguish between appropriate units for area and linear measures

4.2.d. Model volume using cubic units

4.2.e. Use, apply, and select appropriate scales on number lines, graphs, and maps

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. In the real number system, positive rational numbers are represented in multiple equivalent forms. Students can:

1.1.a. Read, write, compare, convert and order positive rational numbers in a variety of forms including proper and improper fractions, mixed numbers, decimals, and percents

1.1.b. Express whole numbers as products of prime factors with exponents and use prime factorization to find the greatest common factor and least common multiple of two numbers

1.1.c. Express the quotient and remainder of a whole number division problem (a/b or a / b) using fractions, terminating decimals, or repeating decimals

1.1.d. Locate positive fractions and decimals on a number line

1.2. Formulate, represent, and use algorithms with positive rational numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Model and compute the addition, subtraction, multiplication and division of positive fractions, decimals, and combinations of fractions and decimals

1.2.b. Solve multi-step word problems involving fractions, decimals and whole numbers

1.2.c. Estimate sums, differences, products and quotients of rational numbers using common fractions, common decimals, and whole numbers

1.2.d. Compare and round positive numbers from thousandths through millions

1.3. Quantities can be expressed and compared using ratios and rates. Students can:

1.3.a. Apply the multiplicative identity to create equivalent fractions and to reduce fractions to simplest form

1.3.b. Express the comparison of two whole number quantities using differences, part-to-part ratios, and part-to-whole ratios in real contexts, including investing and saving (PFL)

1.3.c. Compute unit rates in real-world situations involving mixtures, concentrations, and distance-time relationships

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Patterns can be described using words, tables, and graphs. Students can:

2.1.a. Extend the pattern and describe the rule for arithmetic and geometric sequences

2.1.b. Model linear situations using tables and graphs, and convert between these two representations

2.1.c. Given a linear equation, substitute non-negative input values to create a table and graph coordinate points in the first quadrant

2.2. Variables are used to represent unknown quantities. Students can:

2.2.a. Describe patterns by using words and variables with mathematical symbols

2.2.b. Evaluate expressions by substituting whole number values for variables

CO.3. Data Analysis, Statistics, and Probability

3.1. Questions can be answered by collecting and analyzing data and data displays. Students can:

3.1.a. Formulate questions for populations larger than the classroom

3.1.b. Recognize that a sample may not represent a population accurately

3.1.c. Recognize bias in surveys

3.1.d. Utilize appropriate techniques to design a random sample

3.1.e. Recognize the use of deceptive scales on a graph that make differences look much larger than they are, or the use of pictographs with areas that are proportioned incorrectly

3.2. Mathematical models are used to determine probability. Students can:

3.2.a. Determine probabilities through experiments or simulations

3.2.b. Express the probability of an event using fractions, decimals, and percents

3.2.c. Make a table, tree diagram or an organized list to determine possible outcomes of two or more compound events

3.2.d. Predict outcomes of events using experimental and theoretical probabilities

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Polygons can be described, classified, and analyzed by their attributes. Students can:

4.1.a. Develop and apply formulas and procedures for finding area of triangles, parallelograms, and trapezoids

4.1.b. Describe properties of polygons up to ten sides using accurate vocabulary and notation

4.1.c. Classify triangles and apply angle and side properties, including the sum of the interior angles

4.1.d. Use accurate geometric notation to describe angles, lines, and segments

4.2. Standard units provide common language for communicating measurements. Students can:

4.2.a. Connect metric prefixes to place value

4.2.b. Measure to the nearest sixteenth of an inch

4.2.c. Select and use appropriate units to accurately measure length, weight, capacity and time in problem-solving situations

4.2.d. Use a protractor to measure angles to the nearest degree

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. In the real number system, rational numbers have a unique location on the number line. Students can:

1.1.a. Read, write, locate on number line, compare and order integers and positive rational numbers

1.1.b. Apply the definition of absolute value with integers, quantifying the distance from zero

1.1.c. Express large and small numbers using scientific notation

1.2. Formulate, represent, and use algorithms with integers and positive rational numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Simplify numeric expressions using the order of operations

1.2.b. Add, subtract, multiply, and divide integers

1.2.c. Use mental math and estimation strategies to solve problems involving percents

1.2.d. Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease (PFL)

1.3. Proportional reasoning involves comparisons and multiplicative relationships among ratios. Students can:

1.3.a. Use ratio relationships to solve for a missing value in a proportion

1.3.b. Model proportional relationships with bar models, ratio tables, and similar figures

1.3.c. Explain the difference between a ratio, rate, and unit rate

1.3.d. Estimate and compute unit cost of consumables (to include unit conversions if necessary) sold in quantity to make purchase decisions based on cost and practicality (PFL)

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Relationships involving the constant rate of change are modeled and solved using linear functions. Students can:

2.1.a. Given a linear situation (including direct variation), identify variables and write an equation in slope-intercept form

2.1.b. Given a linear equation (including direct variation), substitute input values to create a table and graph coordinate points in all four quadrants

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays and summary statistics with one-variable data condense the information in data sets into usable knowledge. Students can:

3.1.a. Distinguish between median as middle number and mean as balance point for an ordered set of data

3.1.b. Use Mean Absolute Deviation (MAD) to analyze the spread of a set of data

3.1.c. Construct and interpret dot plots, histograms, stem-and-leaf plots, and circle graphs

3.1.d. Construct and interpret a box plot using the five-number summary and identify the interquartile range (IQR) for a set of data

3.1.e. Compare sets of data using shape (skewed, normal, uniform), with appropriate measures of central tendency (mean, median, mode), and appropriate measures of spread (range, IQR, MAD)

3.1.f. Given a frequency table, calculate relative frequencies

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Objects in space and their parts and attributes can be measured and analyzed. Students can:

4.1.a. Develop and apply formulas and procedures for the surface area and volume of right cylinders and right prisms

4.1.b. Develop and apply formulas and procedures for area of regular polygons, circumference and area of circles, and area of composite figures

4.1.c. Identify and construct two-dimensional nets of prisms and cylinders

4.2. Proportional reasoning is used to make indirect measurements. Students can:

4.2.a. Describe the relationship between the circumference and diameter of a circle

4.2.b. Read and interpret scales on maps

4.2.c. Use proportions to convert from one set of units to another within customary and metric systems using standard units of measure for length, weight, capacity and time

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. In the real number system, rational and irrational numbers are in one to one correspondence to points on the number line. Students can:

1.1.a. Compare and order sets of integers and rational numbers that are expressed as fractions, decimals, or percents

1.1.b. Given a whole number from 0 - 100, determine whether it is a perfect square or find the two consecutive whole numbers between which its square root lies

1.1.c. Approximate the location of square roots between two whole numbers on a number line

1.2. Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Add, subtract, multiply and divide rational numbers including integers, positive and negative fractions and decimals

1.2.b. Apply computational methods to solve multi-step application problems involving percents and rational numbers

1.2.c. Analyze how credit and debt impact personal financial goals (PFL)

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Linear functions model situations with a constant rate of change and can be represented algebraically, graphically, and using tables. Students can:

2.1.a. Convert from one representation of a linear function to another, including situations, tables, equations (slope-intercept form), and graphs

2.1.b. Use representations of linear functions to analyze situations and solve problems

2.1.c. Identify the dependent and independent variable in real-world situations

2.1.d. Identify and interpret the slope (rate of change) and y-intercept in graphs, in tables, and from equations in slope-intercept form

2.1.e. Model and graph two linear equations in slope-intercept form on the same coordinate plane and interpret the point of intersection as the solution to the system of equations

2.2. Properties of algebra, equality, and inequality are used to solve linear equations and inequalities. Students can:

2.2.a. Use the distributive, associative, and commutative properties to simplify algebraic expressions

2.2.b. Solve one-variable equations including those involving multiple steps, rational numbers, variables on both sides, and the distributive property

2.2.c. Solve inequalities in one variable including negative coefficients and graph the solution on a number line

2.2.d. Represent the distributive property in a variety of ways including numerically, geometrically, and algebraically

2.3. Graphs and tables can be used to distinguish between linear and nonlinear functions. Students can:

2.3.a. Given a table or graph determine if the function is linear

2.3.b. Explain the properties of linear functions in tables and graphs

CO.3. Data Analysis, Statistics, and Probability

3.1. Visual displays and summary statistics of two-variable data condense the information in data sets into usable knowledge. Students can:

3.1.a. Given a scatter plot, calculate quadrant count ratio to quantify the magnitude and strength of the association between two variables for numeric data as positive, negative, or no correlation

3.1.b. Given a scatter plot suggesting a linear relationship, draw a line of fit to make predictions

3.1.c. Use time series plots (line graphs) to analyze the trend of a set of data over time

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Objects in the plane and their parts and attributes can be analyzed. Students can:

4.1.a. Classify quadrilaterals and apply angle and side properties, including the sum of the interior angles

4.1.b. Apply properties of complementary, supplementary, and vertical angle relationships

4.1.c. Apply properties of parallel lines including corresponding angles and alternate interior angles

4.2. Direct and indirect measurements can be used to describe and make comparisons. Students can:

4.2.a. Use properties of similar triangles to find unknown lengths

4.2.b. Use the Pythagorean Theorem to find unknown lengths in right triangles

4.2.c. Use proportional reasoning to estimate distance, weight, and capacity

4.2.d. Use proportional reasoning to convert among measures including dimensional analysis

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The complex number system includes real numbers and imaginary numbers. Students can:

1.1.a. Show that between any two rational numbers there are an infinite number of rational numbers, and that between any two irrational numbers there are also an infinite number of irrational numbers

1.1.b. Express the square root of a negative number using imaginary numbers

1.2. Formulate, represent, and use algorithms with real numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Use appropriate computation methods that encompass estimation and calculation

1.2.b. Use technology to perform operations (addition, subtraction, multiplication, and division) on numbers written in scientific notation

1.2.c. Describe factors affecting take-home pay and calculate the impact (PFL)

1.2.d. Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future (PFL)

1.3. Systematic counting techniques are used to describe and solve problems. Students can:

1.3.a. Use combinatorics (Fundamental Counting Principle, permutations and combinations) to solve problems in real-world contexts

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables. Students can:

2.1.a. Determine when a relation is a function using a table, a graph, or an equation

2.1.b. Demonstrate the relationship between all representations of linear functions using point-slope, slope-intercept, and standard form of a line

2.1.c. Represent linear, quadratic, absolute value, power, exponential, logarithmic, rational, trigonometric (sine and cosine), and step functions in a table, graph, and equation and convert from one representation to another

2.1.d. Determine the inverse (expressed graphically or in tabular form) of a function from a graph or table

2.1.e. Categorize sequences as arithmetic, geometric, or neither and develop formulas for the general terms related to arithmetic and geometric sequences

2.2. Graphs and tables are used to describe the qualitative behavior of common types of functions. Students can:

2.2.a. Evaluate a function at a given point in its domain given an equation (including function notation), a table, and a graph

2.2.b. Identify the domain and range of a function given an equation (including function notation), a table, and a graph

2.2.c. Identify intercepts, zeros (or roots), maxima, minima, and intervals of increase and decrease, and asymptotes of a function given an equation (including function notation), a table, and a graph

2.2.d. Make qualitative statements about the rate of change of a function, based on its graph or table

2.3. Parameters influence the shape of the graphs of functions. Students can:

2.3.a. Apply transformations (translation, reflection, dilation) to a parent function, f(x)

2.3.b. Interpret the results of these transformations verbally, graphically, and symbolically

2.4. Expressions, equations, and inequalities can be expressed in multiple, equivalent forms. Students can:

2.4.a. Perform and justify steps in generating equivalent expressions by identifying properties used including the commutative, associative, inverse, identity, and distributive properties

2.4.b. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions including those involving nth roots

2.4.c. Solve equations for one variable in terms of the others

2.5. Solutions to equations, inequalities and systems of equations are found using a variety of tools. Students can:

2.5.a. Find solutions to quadratic and cubic equations and inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula

2.5.b. Find solutions to equations involving power, exponential, rational and radical functions

2.5.c. Solve systems of linear equations and inequalities with two variables

2.6. Quantitative relationships in the real world can be modeled and solved using functions. Students can:

2.6.a. Represent, solve, and interpret problems in various contexts using linear, quadratic, and exponential functions

2.6.b. Represent, solve, and interpret problems involving direct and inverse variations and a combination of direct and inverse variation

2.6.c. Analyze the impact of interest rates on a personal financial plan (PFL)

2.6.d. Evaluate the costs and benefits of credit (PFL)

2.6.e. Analyze various lending sources, services, and financial institutions (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Statistical methods take variability into account, supporting informed decision-making through quantitative studies designed to answer specific questions. Students can:

3.1.a. Formulate appropriate research questions that can be answered with statistical analysis

3.1.b. Determine appropriate data collection methods to answer a research question

3.1.c. Explain how data might be analyzed to provide answers to a research question

3.2. The design of an experiment or sample survey is of critical importance to analyzing the data and drawing conclusions. Students can:

3.2.a. Identify the characteristics of a well-designed and well-conducted survey

3.2.b. Identify the characteristics of a well-designed and well-conducted experiment

3.2.c. Differentiate between the inferences that can be drawn in experiments versus observational studies

3.3. Visual displays and summary statistics condense the information in data sets into usable knowledge. Students can:

3.3.a. Identify and choose appropriate ways to summarize numerical or categorical data using tables, graphical displays, and numerical summary statistics (describing shape, center and spread) and accounting for outliers when appropriate

3.3.b. Define and explain how sampling distributions (developed through simulation) are used to describe the sample-to-sample variability of sample statistics

3.3.c. Describe the relationship between two categorical variables using percents

3.3.d. When the relationship between two numerical variables is reasonably linear, apply the least-squares criterion for line fitting, use Pearson's correlation coefficient as a measure of strength, and interpret the slope and y-intercept in the context of the problem

3.4. Randomness is the foundation for using statistics to draw conclusions when testing a claim or estimating plausible values for a population characteristic. Students can:

3.4.a. Define and explain the meaning of significance (both practical and statistical)

3.4.b. Explain the role of p-values in determining statistical significance

3.4.c. Determine the margin of error associated with an estimate of a population characteristic

3.5. Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of certainty in terms of relative frequency of occurrence. Students can:

3.5.a. Develop simulations that demonstrate probability as a long-run relative frequency

3.5.b. Apply and solve problems using the concepts of independence and conditional probability

3.5.c. Apply and solve problems using the concept of mutually exclusive properties when combining probabilities

3.5.d. Evaluate and interpret probabilities using a normal distribution

3.5.e. Find and interpret the expected value and standard deviation of a discrete random variable X

3.5.f. Analyze the cost of insurance as a method to offset the risk of a situation (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Attributes of two- and three-dimensional objects are measurable and can be quantified. Students can:

4.1.a. Calculate (or estimate when appropriate) the perimeter and area of a two-dimensional irregular shape

4.1.b. Justify, interpret, and apply the use of formulas for the surface area, and volume of cones, pyramids, and spheres including real-world situations

4.1.c. Solve for unknown quantities in relationships involving perimeter, area, surface area, and volume

4.1.d. Apply the effect of dimensional change, utilizing appropriate units and scales in problem-solving situations involving perimeter, area, and volume

4.2. Objects in the plane and their parts, attributes, and measurements can be analyzed deductively. Students can:

4.2.a. Classify polygons according to their similarities and differences

4.2.b. Solve for unknown attributes of geometric shapes based on their congruence, similarity, or symmetry

4.2.c. Know and apply properties of angles including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve problems. Justify the results using two-column proofs, paragraph proofs, flow charts, or illustrations

4.2.d. Develop conjectures and solve problems about geometric figures including definitions and properties (congruence, similarity, and symmetry). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, or illustrations

4.3. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. Students can:

4.3.a. Make conjectures involving two-dimensional objects represented with Cartesian coordinates. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.b. Represent transformations (reflection, translation, rotation, and dilation) using Cartesian coordinates

4.3.c. Develop arguments to establish what remains invariant and what changes after a transformation (reflection, translation, rotation, and dilations). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.d. Using construction tools, including technology, make conjectures about relationships among properties of shapes in the plane including those formed through transformation. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.4. Right triangles are central to geometry and its applications. Students can:

4.4.a. Apply right triangle trigonometry (sine, cosine, and tangent) to find indirect measures of lengths and angles

4.4.b. Apply the Pythagorean theorem and its converse to solve real-world problems

4.4.c. Determine the midpoint of a line segment and the distance between two points in the Cartesian coordinate plane

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The complex number system includes real numbers and imaginary numbers. Students can:

1.1.a. Show that between any two rational numbers there are an infinite number of rational numbers, and that between any two irrational numbers there are also an infinite number of irrational numbers

1.1.b. Express the square root of a negative number using imaginary numbers

1.2. Formulate, represent, and use algorithms with real numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Use appropriate computation methods that encompass estimation and calculation

1.2.b. Use technology to perform operations (addition, subtraction, multiplication, and division) on numbers written in scientific notation

1.2.c. Describe factors affecting take-home pay and calculate the impact (PFL)

1.2.d. Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future (PFL)

1.3. Systematic counting techniques are used to describe and solve problems. Students can:

1.3.a. Use combinatorics (Fundamental Counting Principle, permutations and combinations) to solve problems in real-world contexts

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables. Students can:

2.1.a. Determine when a relation is a function using a table, a graph, or an equation

2.1.b. Demonstrate the relationship between all representations of linear functions using point-slope, slope-intercept, and standard form of a line

2.1.c. Represent linear, quadratic, absolute value, power, exponential, logarithmic, rational, trigonometric (sine and cosine), and step functions in a table, graph, and equation and convert from one representation to another

2.1.d. Determine the inverse (expressed graphically or in tabular form) of a function from a graph or table

2.1.e. Categorize sequences as arithmetic, geometric, or neither and develop formulas for the general terms related to arithmetic and geometric sequences

2.2. Graphs and tables are used to describe the qualitative behavior of common types of functions. Students can:

2.2.a. Evaluate a function at a given point in its domain given an equation (including function notation), a table, and a graph

2.2.b. Identify the domain and range of a function given an equation (including function notation), a table, and a graph

2.2.c. Identify intercepts, zeros (or roots), maxima, minima, and intervals of increase and decrease, and asymptotes of a function given an equation (including function notation), a table, and a graph

2.2.d. Make qualitative statements about the rate of change of a function, based on its graph or table

2.3. Parameters influence the shape of the graphs of functions. Students can:

2.3.a. Apply transformations (translation, reflection, dilation) to a parent function, f(x)

2.3.b. Interpret the results of these transformations verbally, graphically, and symbolically

2.4. Expressions, equations, and inequalities can be expressed in multiple, equivalent forms. Students can:

2.4.a. Perform and justify steps in generating equivalent expressions by identifying properties used including the commutative, associative, inverse, identity, and distributive properties

2.4.b. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions including those involving nth roots

2.4.c. Solve equations for one variable in terms of the others

2.5. Solutions to equations, inequalities and systems of equations are found using a variety of tools. Students can:

2.5.a. Find solutions to quadratic and cubic equations and inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula

2.5.b. Find solutions to equations involving power, exponential, rational and radical functions

2.5.c. Solve systems of linear equations and inequalities with two variables

2.6. Quantitative relationships in the real world can be modeled and solved using functions. Students can:

2.6.a. Represent, solve, and interpret problems in various contexts using linear, quadratic, and exponential functions

2.6.b. Represent, solve, and interpret problems involving direct and inverse variations and a combination of direct and inverse variation

2.6.c. Analyze the impact of interest rates on a personal financial plan (PFL)

2.6.d. Evaluate the costs and benefits of credit (PFL)

2.6.e. Analyze various lending sources, services, and financial institutions (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Statistical methods take variability into account, supporting informed decision-making through quantitative studies designed to answer specific questions. Students can:

3.1.a. Formulate appropriate research questions that can be answered with statistical analysis

3.1.b. Determine appropriate data collection methods to answer a research question

3.1.c. Explain how data might be analyzed to provide answers to a research question

3.2. The design of an experiment or sample survey is of critical importance to analyzing the data and drawing conclusions. Students can:

3.2.a. Identify the characteristics of a well-designed and well-conducted survey

3.2.b. Identify the characteristics of a well-designed and well-conducted experiment

3.2.c. Differentiate between the inferences that can be drawn in experiments versus observational studies

3.3. Visual displays and summary statistics condense the information in data sets into usable knowledge. Students can:

3.3.a. Identify and choose appropriate ways to summarize numerical or categorical data using tables, graphical displays, and numerical summary statistics (describing shape, center and spread) and accounting for outliers when appropriate

3.3.b. Define and explain how sampling distributions (developed through simulation) are used to describe the sample-to-sample variability of sample statistics

3.3.c. Describe the relationship between two categorical variables using percents

3.3.d. When the relationship between two numerical variables is reasonably linear, apply the least-squares criterion for line fitting, use Pearson's correlation coefficient as a measure of strength, and interpret the slope and y-intercept in the context of the problem

3.4. Randomness is the foundation for using statistics to draw conclusions when testing a claim or estimating plausible values for a population characteristic. Students can:

3.4.a. Define and explain the meaning of significance (both practical and statistical)

3.4.b. Explain the role of p-values in determining statistical significance

3.4.c. Determine the margin of error associated with an estimate of a population characteristic

3.5. Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of certainty in terms of relative frequency of occurrence. Students can:

3.5.a. Develop simulations that demonstrate probability as a long-run relative frequency

3.5.b. Apply and solve problems using the concepts of independence and conditional probability

3.5.c. Apply and solve problems using the concept of mutually exclusive properties when combining probabilities

3.5.d. Evaluate and interpret probabilities using a normal distribution

3.5.e. Find and interpret the expected value and standard deviation of a discrete random variable X

3.5.f. Analyze the cost of insurance as a method to offset the risk of a situation (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Attributes of two- and three-dimensional objects are measurable and can be quantified. Students can:

4.1.a. Calculate (or estimate when appropriate) the perimeter and area of a two-dimensional irregular shape

4.1.b. Justify, interpret, and apply the use of formulas for the surface area, and volume of cones, pyramids, and spheres including real-world situations

4.1.c. Solve for unknown quantities in relationships involving perimeter, area, surface area, and volume

4.1.d. Apply the effect of dimensional change, utilizing appropriate units and scales in problem-solving situations involving perimeter, area, and volume

4.2. Objects in the plane and their parts, attributes, and measurements can be analyzed deductively. Students can:

4.2.a. Classify polygons according to their similarities and differences

4.2.b. Solve for unknown attributes of geometric shapes based on their congruence, similarity, or symmetry

4.2.c. Know and apply properties of angles including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve problems. Justify the results using two-column proofs, paragraph proofs, flow charts, or illustrations

4.2.d. Develop conjectures and solve problems about geometric figures including definitions and properties (congruence, similarity, and symmetry). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, or illustrations

4.3. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. Students can:

4.3.a. Make conjectures involving two-dimensional objects represented with Cartesian coordinates. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.b. Represent transformations (reflection, translation, rotation, and dilation) using Cartesian coordinates

4.3.c. Develop arguments to establish what remains invariant and what changes after a transformation (reflection, translation, rotation, and dilations). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.d. Using construction tools, including technology, make conjectures about relationships among properties of shapes in the plane including those formed through transformation. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.4. Right triangles are central to geometry and its applications. Students can:

4.4.a. Apply right triangle trigonometry (sine, cosine, and tangent) to find indirect measures of lengths and angles

4.4.b. Apply the Pythagorean theorem and its converse to solve real-world problems

4.4.c. Determine the midpoint of a line segment and the distance between two points in the Cartesian coordinate plane

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The complex number system includes real numbers and imaginary numbers. Students can:

1.1.a. Show that between any two rational numbers there are an infinite number of rational numbers, and that between any two irrational numbers there are also an infinite number of irrational numbers

1.1.b. Express the square root of a negative number using imaginary numbers

1.2. Formulate, represent, and use algorithms with real numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Use appropriate computation methods that encompass estimation and calculation

1.2.b. Use technology to perform operations (addition, subtraction, multiplication, and division) on numbers written in scientific notation

1.2.c. Describe factors affecting take-home pay and calculate the impact (PFL)

1.2.d. Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future (PFL)

1.3. Systematic counting techniques are used to describe and solve problems. Students can:

1.3.a. Use combinatorics (Fundamental Counting Principle, permutations and combinations) to solve problems in real-world contexts

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables. Students can:

2.1.a. Determine when a relation is a function using a table, a graph, or an equation

2.1.b. Demonstrate the relationship between all representations of linear functions using point-slope, slope-intercept, and standard form of a line

2.1.c. Represent linear, quadratic, absolute value, power, exponential, logarithmic, rational, trigonometric (sine and cosine), and step functions in a table, graph, and equation and convert from one representation to another

2.1.d. Determine the inverse (expressed graphically or in tabular form) of a function from a graph or table

2.1.e. Categorize sequences as arithmetic, geometric, or neither and develop formulas for the general terms related to arithmetic and geometric sequences

2.2. Graphs and tables are used to describe the qualitative behavior of common types of functions. Students can:

2.2.a. Evaluate a function at a given point in its domain given an equation (including function notation), a table, and a graph

2.2.b. Identify the domain and range of a function given an equation (including function notation), a table, and a graph

2.2.c. Identify intercepts, zeros (or roots), maxima, minima, and intervals of increase and decrease, and asymptotes of a function given an equation (including function notation), a table, and a graph

2.2.d. Make qualitative statements about the rate of change of a function, based on its graph or table

2.3. Parameters influence the shape of the graphs of functions. Students can:

2.3.a. Apply transformations (translation, reflection, dilation) to a parent function, f(x)

2.3.b. Interpret the results of these transformations verbally, graphically, and symbolically

2.4. Expressions, equations, and inequalities can be expressed in multiple, equivalent forms. Students can:

2.4.a. Perform and justify steps in generating equivalent expressions by identifying properties used including the commutative, associative, inverse, identity, and distributive properties

2.4.b. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions including those involving nth roots

2.4.c. Solve equations for one variable in terms of the others

2.5. Solutions to equations, inequalities and systems of equations are found using a variety of tools. Students can:

2.5.a. Find solutions to quadratic and cubic equations and inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula

2.5.b. Find solutions to equations involving power, exponential, rational and radical functions

2.5.c. Solve systems of linear equations and inequalities with two variables

2.6. Quantitative relationships in the real world can be modeled and solved using functions. Students can:

2.6.a. Represent, solve, and interpret problems in various contexts using linear, quadratic, and exponential functions

2.6.b. Represent, solve, and interpret problems involving direct and inverse variations and a combination of direct and inverse variation

2.6.c. Analyze the impact of interest rates on a personal financial plan (PFL)

2.6.d. Evaluate the costs and benefits of credit (PFL)

2.6.e. Analyze various lending sources, services, and financial institutions (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Statistical methods take variability into account, supporting informed decision-making through quantitative studies designed to answer specific questions. Students can:

3.1.a. Formulate appropriate research questions that can be answered with statistical analysis

3.1.b. Determine appropriate data collection methods to answer a research question

3.1.c. Explain how data might be analyzed to provide answers to a research question

3.2. The design of an experiment or sample survey is of critical importance to analyzing the data and drawing conclusions. Students can:

3.2.a. Identify the characteristics of a well-designed and well-conducted survey

3.2.b. Identify the characteristics of a well-designed and well-conducted experiment

3.2.c. Differentiate between the inferences that can be drawn in experiments versus observational studies

3.3. Visual displays and summary statistics condense the information in data sets into usable knowledge. Students can:

3.3.a. Identify and choose appropriate ways to summarize numerical or categorical data using tables, graphical displays, and numerical summary statistics (describing shape, center and spread) and accounting for outliers when appropriate

3.3.b. Define and explain how sampling distributions (developed through simulation) are used to describe the sample-to-sample variability of sample statistics

3.3.c. Describe the relationship between two categorical variables using percents

3.3.d. When the relationship between two numerical variables is reasonably linear, apply the least-squares criterion for line fitting, use Pearson's correlation coefficient as a measure of strength, and interpret the slope and y-intercept in the context of the problem

3.4. Randomness is the foundation for using statistics to draw conclusions when testing a claim or estimating plausible values for a population characteristic. Students can:

3.4.a. Define and explain the meaning of significance (both practical and statistical)

3.4.b. Explain the role of p-values in determining statistical significance

3.4.c. Determine the margin of error associated with an estimate of a population characteristic

3.5. Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of certainty in terms of relative frequency of occurrence. Students can:

3.5.a. Develop simulations that demonstrate probability as a long-run relative frequency

3.5.b. Apply and solve problems using the concepts of independence and conditional probability

3.5.c. Apply and solve problems using the concept of mutually exclusive properties when combining probabilities

3.5.d. Evaluate and interpret probabilities using a normal distribution

3.5.e. Find and interpret the expected value and standard deviation of a discrete random variable X

3.5.f. Analyze the cost of insurance as a method to offset the risk of a situation (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Attributes of two- and three-dimensional objects are measurable and can be quantified. Students can:

4.1.a. Calculate (or estimate when appropriate) the perimeter and area of a two-dimensional irregular shape

4.1.b. Justify, interpret, and apply the use of formulas for the surface area, and volume of cones, pyramids, and spheres including real-world situations

4.1.c. Solve for unknown quantities in relationships involving perimeter, area, surface area, and volume

4.1.d. Apply the effect of dimensional change, utilizing appropriate units and scales in problem-solving situations involving perimeter, area, and volume

4.2. Objects in the plane and their parts, attributes, and measurements can be analyzed deductively. Students can:

4.2.a. Classify polygons according to their similarities and differences

4.2.b. Solve for unknown attributes of geometric shapes based on their congruence, similarity, or symmetry

4.2.c. Know and apply properties of angles including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve problems. Justify the results using two-column proofs, paragraph proofs, flow charts, or illustrations

4.2.d. Develop conjectures and solve problems about geometric figures including definitions and properties (congruence, similarity, and symmetry). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, or illustrations

4.3. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. Students can:

4.3.a. Make conjectures involving two-dimensional objects represented with Cartesian coordinates. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.b. Represent transformations (reflection, translation, rotation, and dilation) using Cartesian coordinates

4.3.c. Develop arguments to establish what remains invariant and what changes after a transformation (reflection, translation, rotation, and dilations). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.d. Using construction tools, including technology, make conjectures about relationships among properties of shapes in the plane including those formed through transformation. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.4. Right triangles are central to geometry and its applications. Students can:

4.4.a. Apply right triangle trigonometry (sine, cosine, and tangent) to find indirect measures of lengths and angles

4.4.b. Apply the Pythagorean theorem and its converse to solve real-world problems

4.4.c. Determine the midpoint of a line segment and the distance between two points in the Cartesian coordinate plane

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

CO.1. Number Sense, Properties, and Operations

1.1. The complex number system includes real numbers and imaginary numbers. Students can:

1.1.a. Show that between any two rational numbers there are an infinite number of rational numbers, and that between any two irrational numbers there are also an infinite number of irrational numbers

1.1.b. Express the square root of a negative number using imaginary numbers

1.2. Formulate, represent, and use algorithms with real numbers flexibly, accurately, and efficiently. Students can:

1.2.a. Use appropriate computation methods that encompass estimation and calculation

1.2.b. Use technology to perform operations (addition, subtraction, multiplication, and division) on numbers written in scientific notation

1.2.c. Describe factors affecting take-home pay and calculate the impact (PFL)

1.2.d. Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future (PFL)

1.3. Systematic counting techniques are used to describe and solve problems. Students can:

1.3.a. Use combinatorics (Fundamental Counting Principle, permutations and combinations) to solve problems in real-world contexts

CO.2. Patterns, Functions, and Algebraic Structures

2.1. Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables. Students can:

2.1.a. Determine when a relation is a function using a table, a graph, or an equation

2.1.b. Demonstrate the relationship between all representations of linear functions using point-slope, slope-intercept, and standard form of a line

2.1.c. Represent linear, quadratic, absolute value, power, exponential, logarithmic, rational, trigonometric (sine and cosine), and step functions in a table, graph, and equation and convert from one representation to another

2.1.d. Determine the inverse (expressed graphically or in tabular form) of a function from a graph or table

2.1.e. Categorize sequences as arithmetic, geometric, or neither and develop formulas for the general terms related to arithmetic and geometric sequences

2.2. Graphs and tables are used to describe the qualitative behavior of common types of functions. Students can:

2.2.a. Evaluate a function at a given point in its domain given an equation (including function notation), a table, and a graph

2.2.b. Identify the domain and range of a function given an equation (including function notation), a table, and a graph

2.2.c. Identify intercepts, zeros (or roots), maxima, minima, and intervals of increase and decrease, and asymptotes of a function given an equation (including function notation), a table, and a graph

2.2.d. Make qualitative statements about the rate of change of a function, based on its graph or table

2.3. Parameters influence the shape of the graphs of functions. Students can:

2.3.a. Apply transformations (translation, reflection, dilation) to a parent function, f(x)

2.3.b. Interpret the results of these transformations verbally, graphically, and symbolically

2.4. Expressions, equations, and inequalities can be expressed in multiple, equivalent forms. Students can:

2.4.a. Perform and justify steps in generating equivalent expressions by identifying properties used including the commutative, associative, inverse, identity, and distributive properties

2.4.b. Apply the properties of positive and negative rational exponents to generate equivalent algebraic expressions including those involving nth roots

2.4.c. Solve equations for one variable in terms of the others

2.5. Solutions to equations, inequalities and systems of equations are found using a variety of tools. Students can:

2.5.a. Find solutions to quadratic and cubic equations and inequalities by using appropriate algebraic methods such as factoring, completing the square, graphing or using the quadratic formula

2.5.b. Find solutions to equations involving power, exponential, rational and radical functions

2.5.c. Solve systems of linear equations and inequalities with two variables

2.6. Quantitative relationships in the real world can be modeled and solved using functions. Students can:

2.6.a. Represent, solve, and interpret problems in various contexts using linear, quadratic, and exponential functions

2.6.b. Represent, solve, and interpret problems involving direct and inverse variations and a combination of direct and inverse variation

2.6.c. Analyze the impact of interest rates on a personal financial plan (PFL)

2.6.d. Evaluate the costs and benefits of credit (PFL)

2.6.e. Analyze various lending sources, services, and financial institutions (PFL)

CO.3. Data Analysis, Statistics, and Probability

3.1. Statistical methods take variability into account, supporting informed decision-making through quantitative studies designed to answer specific questions. Students can:

3.1.a. Formulate appropriate research questions that can be answered with statistical analysis

3.1.b. Determine appropriate data collection methods to answer a research question

3.1.c. Explain how data might be analyzed to provide answers to a research question

3.2. The design of an experiment or sample survey is of critical importance to analyzing the data and drawing conclusions. Students can:

3.2.a. Identify the characteristics of a well-designed and well-conducted survey

3.2.b. Identify the characteristics of a well-designed and well-conducted experiment

3.2.c. Differentiate between the inferences that can be drawn in experiments versus observational studies

3.3. Visual displays and summary statistics condense the information in data sets into usable knowledge. Students can:

3.3.a. Identify and choose appropriate ways to summarize numerical or categorical data using tables, graphical displays, and numerical summary statistics (describing shape, center and spread) and accounting for outliers when appropriate

3.3.b. Define and explain how sampling distributions (developed through simulation) are used to describe the sample-to-sample variability of sample statistics

3.3.c. Describe the relationship between two categorical variables using percents

3.3.d. When the relationship between two numerical variables is reasonably linear, apply the least-squares criterion for line fitting, use Pearson's correlation coefficient as a measure of strength, and interpret the slope and y-intercept in the context of the problem

3.4. Randomness is the foundation for using statistics to draw conclusions when testing a claim or estimating plausible values for a population characteristic. Students can:

3.4.a. Define and explain the meaning of significance (both practical and statistical)

3.4.b. Explain the role of p-values in determining statistical significance

3.4.c. Determine the margin of error associated with an estimate of a population characteristic

3.5. Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of certainty in terms of relative frequency of occurrence. Students can:

3.5.a. Develop simulations that demonstrate probability as a long-run relative frequency

3.5.b. Apply and solve problems using the concepts of independence and conditional probability

3.5.c. Apply and solve problems using the concept of mutually exclusive properties when combining probabilities

3.5.d. Evaluate and interpret probabilities using a normal distribution

3.5.e. Find and interpret the expected value and standard deviation of a discrete random variable X

3.5.f. Analyze the cost of insurance as a method to offset the risk of a situation (PFL)

CO.4. Shape, Dimension, and Geometric Relationships

4.1. Attributes of two- and three-dimensional objects are measurable and can be quantified. Students can:

4.1.a. Calculate (or estimate when appropriate) the perimeter and area of a two-dimensional irregular shape

4.1.b. Justify, interpret, and apply the use of formulas for the surface area, and volume of cones, pyramids, and spheres including real-world situations

4.1.c. Solve for unknown quantities in relationships involving perimeter, area, surface area, and volume

4.1.d. Apply the effect of dimensional change, utilizing appropriate units and scales in problem-solving situations involving perimeter, area, and volume

4.2. Objects in the plane and their parts, attributes, and measurements can be analyzed deductively. Students can:

4.2.a. Classify polygons according to their similarities and differences

4.2.b. Solve for unknown attributes of geometric shapes based on their congruence, similarity, or symmetry

4.2.c. Know and apply properties of angles including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve problems. Justify the results using two-column proofs, paragraph proofs, flow charts, or illustrations

4.2.d. Develop conjectures and solve problems about geometric figures including definitions and properties (congruence, similarity, and symmetry). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, or illustrations

4.3. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically. Students can:

4.3.a. Make conjectures involving two-dimensional objects represented with Cartesian coordinates. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.b. Represent transformations (reflection, translation, rotation, and dilation) using Cartesian coordinates

4.3.c. Develop arguments to establish what remains invariant and what changes after a transformation (reflection, translation, rotation, and dilations). Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.3.d. Using construction tools, including technology, make conjectures about relationships among properties of shapes in the plane including those formed through transformation. Justify these conjectures using two-column proofs, paragraph proofs, flow charts, and/or illustrations

4.4. Right triangles are central to geometry and its applications. Students can:

4.4.a. Apply right triangle trigonometry (sine, cosine, and tangent) to find indirect measures of lengths and angles

4.4.b. Apply the Pythagorean theorem and its converse to solve real-world problems

4.4.c. Determine the midpoint of a line segment and the distance between two points in the Cartesian coordinate plane

CO.5. Prepared Graduate Competencies in Mathematics: The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.

5.1. Understand the structure and properties of our number system. At the most basic level numbers are abstract symbols that represent real-world quantities

5.2. Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

5.3. Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

5.4. Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

5.5. Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

5.6. Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data

5.7. Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

5.8. Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

5.9. Apply transformation to numbers, shapes, functional representations, and data

5.10. Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics

5.11. Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking

5.12. Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions