# BPS District Mathematics Standards Book

Completion requirements

### Math Domains

#### Number and Operations

####
(CC) Counting and Cardinality

*Learners will understand the relationship between numerical symbols, names, quantities, and counting sequences.*

** Counting Forward**

**MAT-00.NO.CC.01**Count verbally in sequential order by ones and tens to 100, making accurate decuple transitions (ex.89 to 90). Count verbally forward from any given number within 100.**MAT-01.NO.CC.01**Count forward by ones and tens from any given point within 120.**MAT-02.NO.CC.01**Count forward from any given number within 1000.

** Counting Backward**

**MAT-00.NO.CC.02**Count backward from 20 by ones and from a given number within 10.**MAT-01.NO.CC.02**Count backward by ones and tens from any given number within 120.**MAT-02.NO.CC.02**Count backward from any given number within 1000.

** Number Identification and Writing**

**MAT-00.NO.CC.03**Identify and write any given numeral within 20.**MAT-01.NO.CC.03**Represent several objects with a written numeral up to 120.**MAT-02.NO.CC.03**Read and write numbers up to 1,000 using standard, word, and expanded forms.**MAT-03.NO.CC.01**Read and write numbers up to 10,000 using objects or visual representations including standard and expanded forms.**MAT-04.NO.CC.01**Read numbers to the millions place including word, standard and expanded form. Write numbers to the millions place including standard and expanded form.**MAT-05.NO.CC.01**Read, write, and compare decimals to the thousandths including standard and expanded forms.

** Subitizing**

**MAT-00.NO.CC.04**Recognize and verbally label arrangements, without counting, for briefly shown collections up to 10 (e.g., "I saw 5." How do you know?" "I saw 3 and 2, that is 5.").**MAT-01.NO.CC.04**Recognize and verbally label arrangements, without counting, for briefly shown collections up to 20 (e.g., "I saw 16." "How did you know?" "I saw 10 and 6, that is 16.").

** Counting Patterns**

**MAT-00.NO.CC.05**Count and tell how many objects up to 20 are in an arranged pattern or up to 10 objects in a scattered configuration. Represent a quantity of up to 20 with a numeral.**MAT-01.NO.CC.05**Skip count forward and backward by 5s and 10s from multiples and recognize the patterns of up to 10 skip counts.**MAT-02.NO.CC.04**Skip count forward and backward by 2s and 100s and recognize the patterns of skip counts.**MAT-10.DPS.09**Determine the number of outcomes using permutations and combinations in context.**MAT-12.DPS.11**Use permutations and combinations to compute probabilities of compound events and solve problems.

#### (NBT) Number Base Ten K-5

*Learners will understand the place value structure of the base-ten number system and represent, compare, and perform operations with multi-digit whole numbers and decimals.*

#### (NS) Number Sense MS

*Learners will expand their knowledge of the number system to create connections and solve problems within and across concepts.*

** Place Value**

**MAT-00.NO.NBT.01**Compose and decompose numbers from 11 to 19 using a group of ten ones and some more ones using a model, drawing, or equation.**MAT-01.NO.NBT.01**Demonstrate that the two digits of a two-digit number represent a composition of some tens and some ones.**MAT-02.NO.NBT.01**Understand that the three digits of a three-digit number represent a composition of some hundreds, some tens, and some ones.**MAT-04.NO.NBT.01**Understand that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.**MAT-05.NO.NBT.01**Understand that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.**MAT-08.NO.NS.03**Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.

** Compare Numbers and/or Expressions**

**MAT-00.NO.NBT.02**Compare two numbers between 1 and 20 using words greater than, less than, or equal to.**MAT-01.NO.NBT.02**Compare two two-digit numbers using symbols >, <, and =. Justify comparisons based on the number of tens and ones.**MAT-02.NO.NBT.02**Compare two three-digit numbers using symbols >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.**MAT-03.NO.NBT.01**Compare two four-digit numbers using symbols, >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.**MAT-04.NO.NBT.02**Compare two numbers to the millions place and decimals to the hundredths place, using symbols >, <, and =. Justify comparisons based on the value of the digits.**MAT-05.NO.NBT.02**Compare two decimals to thousandths using symbols >, <, and =. Justify comparisons based on the value of the digits.**MAT-06.NO.NS.01**Explain and show the relationship between non-zero rational numbers and their opposites using horizontal and vertical number lines in authentic problems. Use rational numbers to represent quantities in authentic contexts and explain the meaning of 0 in certain situations.**MAT-06.NO.NS.02**Write, interpret, and explain statements of order for rational numbers on a number line and in authentic contexts.**MAT-07.NO.NS.01**Describe the absolute value of a number as its distance from zero on a number line.**MAT-08.NO.NS.01**Compare and classify real numbers within the real number system.**MAT-08.NO.NS.02**Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them on a number line diagram, and estimate the value of irrational expressions involving one operation.**MAT-08.NO.NS.03**Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.**MAT-09.AR.06**Solve linear equations and inequalities (to include compound inequalities) in one variable.**MAT-09.AR.08**Graph the solution set to a two-variable system of linear equations. Create and graph the solution set to a two-variable system of linear inequalities in context.**MAT-09.AR.09**Solve absolute value equations and inequalities in one or two variables.**MAT-12.AR.09**Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

** Rounding Numbers**

**MAT-03.NO.NBT.02**Apply place value understanding to round whole numbers to the nearest 10 or 100.**MAT-04.NO.NBT.03**Apply place value and/or understanding of numbers to round multi-digit whole numbers to any place.**MAT-05.NO.NBT.03**Apply place value understanding to round decimals to any place.**MAT-08.NO.NS.03**Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.

#### (NBT) Number Base Ten K-5

*Learners will understand the place value structure of the base-ten number system and represent, compare, and perform operations with multi-digit whole numbers and decimals.*

#### (O) Operations MS

*Learners will expand their computational fluency to create connections and solve problems within and across concepts.*

** Addition and Subtraction**

**MAT-01.NO.NBT.03**Add within 100 using a two-digit number and a one-digit number. Use concrete models, drawings, and strategies that reflect an understanding of place value.**MAT-02.NO.NBT.03**Add within 100 using place value strategies and/or the relationship between addition and subtraction.**MAT-01.NO.NBT.04**Subtract multiples of 10 within 100 using concrete models, drawings, and strategies that reflect an understanding of place value.**MAT-01.NO.NBT.05**Mentally add or subtract 10 to or from a given two-digit number and explain the reasoning used.**MAT-02.NO.NBT.04**Subtract within 100 using place value strategies and/or the relationship between addition and subtraction.**MAT-02.NO.NBT.05**Mentally add or subtract 10 or 100 to or from a given number between 100 and 900.**MAT-03.NO.NBT.03**Add and subtract within 1000 using place value strategies, algorithms, and/or the relationship between addition and subtraction.**MAT-04.NO.NBT.04**Add and subtract multi-digit whole numbers to the one million place using strategies flexibly, including the algorithm.**MAT-05.NO.NBT.05**Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.**MAT-07.NO.O.01**Add, subtract, multiply, and divide integers using visual models and properties of operations in multi-step authentic and mathematical problems, including authentic problems.**MAT-07.NO.O.02**Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.**MAT-07.NO.O.03**Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.**MAT-08.NO.O.02**Add, subtract, multiply, and divide rational numbers using strategies or procedures.**MAT-09.NO.02**Perform basic operations on simple radical expressions to write a simplified equivalent expression.**MAT-09.AR.11**Add, subtract, and multiply polynomials.**MAT-12.NO.03**Demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.**MAT-12.AR.13**Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.**MAT-12.NO.11**Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.**MAT-12.NO.17**Add and subtract vectors. Represent vector subtraction graphically by connecting the tips of the appropriate order and using the components to perform vector subtraction.**MAT-12.NO.19**Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

** Multiplication and Division**

**MAT-03.NO.NBT.04**Multiply one-digit whole numbers by multiples of 10 within 100.**MAT-04.NO.NBT.05**Multiply a whole number up to four digits by a one-digit whole number and multiply two two digit numbers. Show and justify the calculation using equations, rectangular arrays, and models.**MAT-05.NO.NBT.04**Multiply multi-digit whole numbers using strategies flexibly, including the algorithm.**MAT-05.NO.NBT.07**Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.**MAT-04.NO.NBT.06**Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using place value strategies. Show and justify the calculation by using equations, rectangular arrays, and models.**MAT-05.NO.NBT.05**Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.**MAT-05.NO.NBT.06**Find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors using place value strategies. Show and justify the calculation using equations, rectangular arrays, and/or area models.**MAT-06.NO.O.01**Divide multi-digit whole numbers up to four-digit dividends and two-digit divisors using strategies or procedures.**MAT-07.NO.O.01**Add, subtract, multiply, and divide integers and positive rational numbers using visual models and properties of operations in multi-step problems, including authentic problems.**MAT-07.NO.O.02**Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.**MAT-07.NO.O.03**Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.**MAT-08.NO.O.01**Evaluate mentally the square roots of perfect squares up to 225 and cube roots of perfect cubes up to 1000.**MAT-08.NO.O.02**Add, subtract, multiply, and divide rational numbers using strategies or procedures.**MAT-09.NO.02**Perform basic operations on radicals and simplify radicals to write equivalent expressions.**MAT-09.AR.11**Add, subtract, and multiply polynomials.**MAT-12.NO.03**Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.**MAT-12.AR.13**Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.**MAT-12.NO.11**Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.**MAT-12.NO.19**Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

#### (NF) Fractions K-5

*Learners will understand fractions and equivalency to represent, compare, and perform operations of fractions and decimals.*

#### (NS) Number Sense MS

*Learners will expand their knowledge of the number system to create connections and solve problems within and across concepts.*

** Fractions - Partition Shapes**

**MAT-01.NO.NF.01**Partition circles and rectangles into two and four equal shares using the language halves and fourths.**MAT-02.NO.NF.01**Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the language of halves, thirds, fourths, half of, a third of, and a fourth of.**MAT-02.NO.NF.02**Recognize that identical wholes can be equally divided in different ways.**MAT-02.NO.NF.03**Recognize that partitioning shapes into more equal shares creates smaller shares.**MAT-03.NO.NF.01**Partition two-dimensional figures into equal areas and express the area of each part as a unit fraction of the whole. Describe using the language of sixths, eighths, a sixth of, and an eighth of.

** Fractions**

**MAT-03.NO.NF.02**Represent and understand a fraction as a number on a number line.**MAT-03.NO.NF.03**Represent equivalent fractions using visual representations and number lines.**MAT-03.NO.NF.04**Recognize whole numbers as fractions and express fractions that are equivalent to whole numbers.**MAT-03.NO.NF.05**Compare fractions of the same whole having the same numerators or denominators, using symbols >, <, and = by reasoning about their size. (Fractions should be limited to denominators of 2, 3, 4, 6, and 8 and should not exceed the whole.)**MAT-04.NO.NF.01**Express equivalent fractions with a denominator of 10 and a denominator of 100 to generate a decimal notation.**MAT-04.NO.NF.02**Explain and demonstrate how a mixed number is equivalent to a fraction greater than one and how a fraction greater than one is equivalent to a mixed number using visual fraction models and reasoning strategies (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).**MAT-04.NO.NF.03**Generate equivalent fractions using numerical representations, visual representations, and number lines (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).**MAT-04.NO.NF.04**Demonstrate how equivalent fractions are generated by multiplying a fraction equivalent to 1 or the properties of multiplication (proper and improper fractions limited to 2, 3,4, 5, 6, 8, 10, 12, and 100).**MAT-04.NO.NF.05**Compare and order fractions having unlike numerators or denominators. Record comparisons using symbols >, <, = and justify using a visual fraction model (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).**MAT-05.NO.NF.01**Generate equivalent forms of commonly used fractions and decimals (e.g., halves, fourths, fifths, tenths).**MAT-06.NO.NS.01**Explain and show the relationship between non-zero rational numbers and their opposites using horizontal and vertical number lines in authentic and mathematical problems. Use rational numbers to represent quantities in authentic contexts and explain the meaning of 0 in certain situations.**MAT-06.NO.NS.02**Write, interpret, and explain statements of order for rational numbers on a number line and in authentic contexts.**MAT-07.NO.NS.02**Recognize common fractions and decimal equivalencies up to a denominator of 10. Convert a rational number to a decimal using technology.

#### (NF) Fractions K-5

*Learners will understand fractions and equivalency to represent, compare, and perform operations of fractions and decimals.*

#### (O) Operations MS

*Learners will expand their computational fluency to create connections and solve problems within and across concepts.*

** Adding and Subtracting Fractions**

**MAT-04.NO.NF.06**Solve authentic word problems by adding and subtracting fractions and mixed numbers with like denominators (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).**MAT-05.NO.NF.03**Solve authentic word problems by adding and subtracting fractions and mixed numbers with unlike denominators using a visual fraction model and/or equation.**MAT-06.NO.O.02**Add and subtract fractions and decimals up to the hundredth place, including in authentic problems.**MAT-07.NO.O.02**Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.**MAT-08.NO.O.02**Add, subtract, multiply, and divide rational numbers using strategies or procedures.**MAT-12.NO.04**Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.**MAT-12.AR.05**Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.**MAT-12.NO.06**Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.

** Multiplying and Dividing Fractions**

**MAT-04.NO.NF.07**Solve problems by multiplying fractions and whole numbers using visual fraction models (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).**MAT-05.NO.NF.02**Explain why multiplying a given number by a fraction greater than one results in a product greater than the given number and explain why multiplying a given number by a fraction less than one results in a product smaller than the given number.**MAT-05.NO.NF.04**Solve authentic word problems by multiplying fractions and mixed numbers using visual fraction models and equations.**MAT-06.NO.O.03**Apply multiplication and division of fractions and decimals to solve and interpret problems using visual models, including authentic problems.**MAT-07.NO.O.02**Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.**MAT-08.NO.O.02**Add, subtract, multiply, and divide rational numbers using strategies or procedures.**MAT-09.NO.02**Perform basic operations of simple radical expressions to write a simplified equivalent expression.**MAT-12.NO.03**Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.**MAT-12.NO.02**Perform basic operations on advanced radicals and simplify radicals to write equivalent expressions.**MAT-12.AR.5**Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.**MAT-12.NO.06**Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.

#### (NBT) Number Base Ten K-5

*Learners will understand the place value structure of the base-ten number system and represent, compare, and perform operations with multi-digit whole numbers and decimals.*

#### (EE) Expressions and Equations MS

*Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.*

** Exponents**

**MAT-05.NO.NBT.07**Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

**MAT-06.AR.EE.01**Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.

**MAT-08.AR.EE.01**Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.

**MAT-09.NO.01**Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.

**MAT-09.NO.02**Perform basic operations on simple radical expressions to write a simplified equivalent expression.

**MAT-09.AR.06**Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.

**MAT-09.AR.04**Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with proper labels and scales.

**MAT-09.AR.F.06**Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

**MAT-0.AR.F.08**Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.

**MAT-12.NO.01**Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.

**MAT-12.NO.02**Perform basic operations on advanced radicals and simplify radicals to write equivalent expressions.

**MAT-12.AR.07**Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

**MAT-12.AR.08**Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

**MAT-12.AR.F.03**Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.

**MAT-12.AR.F.06**Apply the inverse relationship between exponents and logarithms to solve problems.

**MAT-12.AR.F.15**Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

** Decimal**

**MAT-04.NO.NF.01**Express equivalent fractions with a denominator of 10 and a denominator of 100 to generate a decimal notation.

**MAT-04.NO.NBT.02**Compare two numbers to the millions place and decimals to the hundredths place, using symbols >, <, and =. Justify comparisons based on the value of the digits.

**MAT-05.NO.CC.01**Read, write, and compare decimals to thousandths including standard form and expanded form.

**MAT-05.NO.NBT.01**Understand that in a multi-digit whole number, a digit in one place represents ten times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

**MAT-05.NO.NBT.02**Compare two decimals to thousandths using symbols >, =, <. Justify comparisons based on the value of the digits.

**MAT-05.NO.NBT.03**Apply place value understanding to round decimals to any place.

**MAT-05.NO.NBT.05**Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.

**MAT-05.NO.NBT.07**Explain patterns in the numbers of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

**MAT-05.NO.NF.01**Generate equivalent forms of commonly used fractions and decimals (e.g., halves, fourths, fifths, tenths).

**MAT-06.NO.O.02**Add and subtract fractions and decimals up to the hundredths place, including authentic problems.

**MAT-07.NO.NS.02**Recognize common fractions and decimal equivalencies up to a denominator of 10. Convert a rational number to a decimal using technology.

**MAT-07.NO.O.03**Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.

**MAT-08.NO.NS.03**Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.

** Unit Size and Scale**

**MAT-06.NO.NS.02**Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.

**MAT-09.NO.03**Choose and interpret the scale and the origin in graphs and data displays.

**MAT-09.NO.04**Define appropriate quantities and units for the purpose of descriptive modeling.

**MAT-09.NO.05**Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

**MAT-09.AR.04**Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

**MAT-10.GM.14**Verify experimentally and justify the properties of dilations given by a center and a scale factor.

**MAT-10.GM.26**Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.

**MAT-12.NO.04**Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.

**MAT-12.NO.05**Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

**MAT-12.AR.08**Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.

** Complex Numbers**

**MAT-12.NO.06**Know there is a complex number i such that i² = -1, and every complex number has the form of a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.

**MAT-12.NO.07**Use the definition i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

**MAT-12.NO.08**Use conjugates to find quotients of complex numbers.

**MAT-12.NO.09**Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.

**MAT-12.AR.11**Solve quadratic equations with real coefficients that have solutions of the form a+bi and a-bi.

**MAT-12.NO.10**Represent complex numbers on the complex plane in rectangular, trigonometric, and polar forms. Find the modulus (absolute value) of a complex number. Explain why the rectangular, trigonometric, and polar forms of a given complex number represent the same number.

**MAT-12.NO.11**Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.

**MAT-12.NO.12**Extend polynomial identities to the complex numbers.

**MAT-12.NO.13**Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.