Core Content Connectors by Common Core State Standards: Mathematics Number and Quantity

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'''Mathematics \| High School—Number and Quantity Overview'''
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='''Mathematics High School—Number and Quantity Overview'''=
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'''The Real Number System'''
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=='''The Real Number System'''==
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'''Extend the properties of exponents to rational exponents'''
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*'''Extend the properties of exponents to rational exponents'''
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'''Use properties of rational and irrational numbers.'''
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*'''Use properties of rational and irrational numbers.'''
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'''Quantities'''
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=='''Quantities'''==
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'''Reason quantitatively and use units to solve problems.'''
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*'''Reason quantitatively and use units to solve problems.'''
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'''The Complex Number System'''
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=='''The Complex Number System'''==
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'''Perform arithmetic operations with complex numbers'''
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*'''Perform arithmetic operations with complex numbers'''
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'''Represent complex numbers and their operations on the complex plane'''
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*'''Represent complex numbers and their operations on the complex plane'''
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'''Use complex numbers in polynomial identities and equations.'''
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*'''Use complex numbers in polynomial identities and equations.'''
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'''Vector and Matrix Quantities '''
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=='''Vector and Matrix Quantities '''==
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'''Represent and model with vector quantities.'''
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*'''Represent and model with vector quantities.'''
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'''Perform operations on vectors.'''
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*'''Perform operations on vectors.'''
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'''Perform operations on matrices and use matrices in applications.'''
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*'''Perform operations on matrices and use matrices in applications.'''
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 +
 
 +
 
 
{|border=1
 
{|border=1
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|width = "50%" style="background-color:#D9D9D9;"|'''The Real Number System'''
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|width = "25%" style="background-color:#D9D9D9;"|'''The Real Number System'''
  
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|width = "50%" style="background-color:#D9D9D9;"|'''N-RN'''
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|width = "75%" style="background-color:#D9D9D9;"|'''N-RN'''
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 4|# Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. ''For example, we define  
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| style="background-color:#FFFFFF;" colspan = 4|1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. ''For example, we define  
 
|-
 
|-
 
| style="background-color:#FFFFFF;"|''CCCs linked to N-RN.1''
 
| style="background-color:#FFFFFF;"|''CCCs linked to N-RN.1''
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Rewrite expressions involving radicals and rational exponents using the properties of exponents.
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| style="background-color:#FFFFFF;" colspan = 2|2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Explain why 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.
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| style="background-color:#FFFFFF;" colspan = 2|3. Explain why 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.
  
 
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{|border=1
 
{|border=1
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|width = "50%" style="background-color:#D9D9D9;"|'''Quantities'''
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|width = "25%" style="background-color:#D9D9D9;"|'''Quantities'''
  
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|width = "50%" style="background-color:#D9D9D9;"|'''N-Q'''
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|width = "75%" style="background-color:#D9D9D9;"|'''N-Q'''
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
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| style="background-color:#FFFFFF;" colspan = 2|1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Define appropriate quantities for the purpose of descriptive modeling.
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| style="background-color:#FFFFFF;" colspan = 2|2. Define appropriate quantities for the purpose of descriptive modeling.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
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| style="background-color:#FFFFFF;" colspan = 2|3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
  
 
|-
 
|-
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{|border=1
 
{|border=1
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|width = "50%" style="background-color:#D9D9D9;"|'''The Complex Number System'''
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|width = "25%" style="background-color:#D9D9D9;"|'''The Complex Number System'''
  
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|width = "50%" style="background-color:#D9D9D9;"|'''N-CN'''
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|width = "75%" style="background-color:#D9D9D9;"|'''N-CN'''
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Know there is a complex number ''i ''such that ''i''² = –1, and every complex number has the form ''a ''+ ''bi ''with ''a ''and ''b ''real.
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| style="background-color:#FFFFFF;" colspan = 2|1. Know there is a complex number ''i ''such that ''i''² = –1, and every complex number has the form ''a ''+ ''bi ''with ''a ''and ''b ''real.
  
 
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Use the relation ''i''² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
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| style="background-color:#FFFFFF;" colspan = 2|2. Use the relation ''i''² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
  
 
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|-
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
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| style="background-color:#FFFFFF;" colspan = 2|3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
  
 
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
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| style="background-color:#FFFFFF;" colspan = 2|4. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
  
 
|-
 
|-
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. ''For example, ''(–1 + √3 i)³ = 8 ''because ''(–1 + √3 i) ''has modulus ''2 ''and argument ''120°.
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| style="background-color:#FFFFFF;" colspan = 2|5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. ''For example, ''(–1 + √3 i)³ = 8 ''because ''(–1 + √3 i) ''has modulus ''2 ''and argument ''120°.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
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| style="background-color:#FFFFFF;" colspan = 2|6. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Solve quadratic equations with real coefficients that have complex solutions.
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| style="background-color:#FFFFFF;" colspan = 2|7. Solve quadratic equations with real coefficients that have complex solutions.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Extend polynomial identities to the complex numbers. ''For example, rewrite x''² + 4 ''as ''(''x ''+ 2''i'')(''x ''– 2''i'').
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| style="background-color:#FFFFFF;" colspan = 2|8. Extend polynomial identities to the complex numbers. ''For example, rewrite x''² + 4 ''as ''(''x ''+ 2''i'')(''x ''– 2''i'').
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
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| style="background-color:#FFFFFF;" colspan = 2|9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., '''''v''''', \|'''''v'''''\|, \|\|'''''v'''''\|\|, ''v'').
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| style="background-color:#FFFFFF;" colspan = 2|1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., '''''v''''', \|'''''v'''''\|, \|\|'''''v'''''\|\|, ''v'').
  
 
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|-
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
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| style="background-color:#FFFFFF;" colspan = 2|2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
  
 
|-
 
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Solve problems involving velocity and other quantities that can be represented by vectors.
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| style="background-color:#FFFFFF;" colspan = 2|3. Solve problems involving velocity and other quantities that can be represented by vectors.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Add and subtract vectors.
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| style="background-color:#FFFFFF;" colspan = 2|4. Add and subtract vectors.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|## Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
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| style="background-color:#FFFFFF;" colspan = 2|a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|## Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
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| style="background-color:#FFFFFF;" colspan = 2|b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|## Understand vector subtraction '''''v '''''– '''''w '''''as '''''v '''''+ (–'''''w'''''), where –'''''w '''''is the additive inverse of '''''w''''', with the same magnitude as '''''w '''''and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
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| style="background-color:#FFFFFF;" colspan = 2|c. Understand vector subtraction '''''v '''''– '''''w '''''as '''''v '''''+ (–'''''w'''''), where –'''''w '''''is the additive inverse of '''''w''''', with the same magnitude as '''''w '''''and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Multiply a vector by a scalar.
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| style="background-color:#FFFFFF;" colspan = 2|5. Multiply a vector by a scalar.
  
 
|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as ''c''(''v''x, ''v''y) = (''cv''x, ''cv''y).
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| style="background-color:#FFFFFF;" colspan = 2|a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as ''c''(''v''x, ''v''y) = (''cv''x, ''cv''y).
  
 
|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Compute the magnitude of a scalar multiple ''c'''v '''''using \|\|''c'''v'''''\|\| = \|''c''\|''v''. Compute the direction of ''c'''v '''''knowing that when \|''c''\|''v ''≠ 0, the direction of ''c'''v '''''is either along '''''v '''''(for ''c ''> 0) or against '''''v '''''(for ''c ''< 0).
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| style="background-color:#FFFFFF;" colspan = 2|b. Compute the magnitude of a scalar multiple ''c'''v '''''using \|\|''c'''v'''''\|\| = \|''c''\|''v''. Compute the direction of ''c'''v '''''knowing that when \|''c''\|''v ''≠ 0, the direction of ''c'''v '''''is either along '''''v '''''(for ''c ''> 0) or against '''''v '''''(for ''c ''< 0).
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
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| style="background-color:#FFFFFF;" colspan = 2|6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  
 
|-
 
|-
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
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| style="background-color:#FFFFFF;" colspan = 2|7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Add, subtract, and multiply matrices of appropriate dimensions.
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| style="background-color:#FFFFFF;" colspan = 2|8. Add, subtract, and multiply matrices of appropriate dimensions.
  
 
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|-
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
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| style="background-color:#FFFFFF;" colspan = 2|9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
  
 
|-
 
|-
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
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| style="background-color:#FFFFFF;" colspan = 2|10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
  
 
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|-
 
|-
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| style="background-color:#FFFFFF;" colspan = 2|# Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
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| style="background-color:#FFFFFF;" colspan = 2|11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
  
 
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|-
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| style="background-color:#FFFFFF;" colspan = 2|# Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
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| style="background-color:#FFFFFF;" colspan = 2|12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
  
 
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Revision as of 13:51, 7 October 2013

Contents

Mathematics High School—Number and Quantity Overview

The Real Number System

  • Extend the properties of exponents to rational exponents
  • Use properties of rational and irrational numbers.

Quantities

  • Reason quantitatively and use units to solve problems.

The Complex Number System

  • Perform arithmetic operations with complex numbers
  • Represent complex numbers and their operations on the complex plane
  • Use complex numbers in polynomial identities and equations.

Vector and Matrix Quantities

  • Represent and model with vector quantities.
  • Perform operations on vectors.
  • Perform operations on matrices and use matrices in applications.


The Real Number System N-RN
Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define
CCCs linked to N-RN.1 None
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
CCCs linked to N-RN.2 HS.NO.1a1 Simplify expressions that include exponents.
HS.NO.1a2 Explain the influence of an exponent on the location of a decimal point in a given number.
HS.NO1a3 Convert a number expressed in scientific notation.
H.NO.2c2 Rewrite expressions that include rational exponents.
Use properties of rational and irrational numbers.
3. Explain why 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.
CCCs linked to N-RN.3 H.NO.2b1 Explain the pattern for the sum or product for combinations of rational and irrational numbers.
Quantities N-Q
Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
CCCs linked to N-Q.1 H.ME.1a1 Determine the necessary unit(s) to use to solve real-world problems.
H.ME.1a2 Solve real-world problems involving units of measurement
2. Define appropriate quantities for the purpose of descriptive modeling.
CCCs linked to N-Q.2 None
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
CCCs linked to N-Q.3 None


The Complex Number System N-CN
Perform arithmetic operations with complex numbers.
1. 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.
CCCs linked to N-CN.1 None
2. Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCCs linked to N-CN.2 None
3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
CCCs linked to N-CN.3 None
Represent complex numbers and their operations on the complex plane.
4. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
CCCs linked to N-CN.4 None
5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)³ = 8 because (–1 + √3 i) has modulus 2 and argument 120°.
CCCs linked to N-CN.5 None
6. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
CCCs linked to N-CN.6 None
Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
CCCs linked to N-CN.7 None
8. Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).
CCCs linked to N-CN.8 None
9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
CCCs linked to N-CN.9 None
Vector and Matrix Quantities N-VM
Represent and model with vector quantities.
1. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, \|v\|, \|\|v\|\|, v).
CCCs linked to N-VM.1 None
2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCCs linked to N-VM.2 None
3. Solve problems involving velocity and other quantities that can be represented by vectors.
CCCs linked to N-VM.3 None
Perform operations on vectors.
4. Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
CCCs linked to N-VM.4 None
5. Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
b. Compute the magnitude of a scalar multiple cv using \|\|cv\|\| = \|c\|v. Compute the direction of cv knowing that when \|c\|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
CCCs linked to N-VM.5 None
Perform operations on matrices and use matrices in applications.
6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
CCCs linked to N-VM.6 None
7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
CCCs linked to N-VM.7 None
8. Add, subtract, and multiply matrices of appropriate dimensions.
CCCs linked to N-VM.8 None
9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
CCCs linked to N-VM.9 None
10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
CCCs linked to N-VM.10 None


11. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
CCCs linked to N-VM.11 None
12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area
'CCCs linked to N-VM.12' None
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