High School – Complex Number System
Essential Questions:
1. Why do we use numbers, what are their properties, and how does our number system function?
2. Why do we use estimation and when is it appropriate?
3. What makes a strategy effective and efficient and the solution reasonable?
4. How do numbers relate and compare to one another?
Essential Vocabulary: conjugate, complex number, moduli, complex plane, rectangular form, polar form, conjugation, quadratic equations, real
coefficients, complex solutions, polynomial identities, Fundamental Theorem of Algebra HS.N-CN.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.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. complex number and i.
1. that all numbers can be written in complex 1. identify a and b in a complex number.
2. tell the value of i.
form.
2. that a is the real part of the number and b
is the coefficient of the imaginary part of a
complex number.
HS.N-CN.2: Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex
numbers.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. add, subtract and multiply complex
1. that the properties of the real number
1. commutative, associative and distributive
numbers.
system, commutative, associative and
properties
distributive properties also apply to
complex numbers.
HS.N-CN.3: (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. conjugate
1. that conjugates can be used to find moduli 1. find the conjugate of a complex number.
2. complex number
and quotients of complex.
2. find moduli and quotients of complex
3. moduli
numbers.
4. quotient
HS.N-CN.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that rectangular and polar forms of a given 1. represent complex numbers on the
1. complex numbers
complex plane in rectangular and polar
complex number represent the same
2. complex plane
form.
number.
3. rectangular and polar form
2. explain the relationship between the
rectangular and polar form of a complex
number.
HS.N-CN.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
8 because 1
Enduring Understandings
Students will understand…
Students will know…
1. that computation and conjugation of
1. conjugation
complex numbers can be represented on
2. addition
the complex plane.
3. subtraction
4. multiplication
5. complex numbers
6. complex plane
7. properties
8. computation
√3
has modulus 2 and argument 120°.
Students will be able to…
1. represent and use properties of addition,
subtraction, multiplication, and
conjugation of complex numbers on the
complex plane.
HS.N-CN.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. calculate the distance between numbers in
1. that distance between numbers in the
1. distance
the complex plane.
complex plan can be calculated as the
2. complex plane
2. calculate the midpoint of a segment.
modulus of the difference.
3. midpoint
4. average
5. modulus
6. endpoints
HS.N-CN.7: Solve quadratic equations with real coefficients that have complex solutions.
Students will be able to…
Students will know…
Students will understand…
1. quadratic formula
1. that when using the quadratic formula you 1. solve a quadratic equation with real
coefficients using the quadratic formula
can arrive at either real or complex
which results in complex solutions.
solutions.
HS.N-CN.8: (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. extend polynomial identities to the
1. that complex numbers can be extended
1. polynomial identities
complex numbers.
from polynomial identities.
2. complex numbers
HS.N-CN.9: (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that the Fundamental Theorem of Algebra 1. show how the Fundamental Theorem is
1. Fundamental Theorem of Algebra
true for quadratic polynomials.
is true for quadratic polynomials.
2. quadratic polynomials
High School - Quantities
Essential Questions:
1. Why do we use numbers, what are their properties, and how does our number system function?
2. Why do we use estimation and when is it appropriate?
3. What makes a strategy effective and efficient and the solution reasonable?
4. How do numbers relate and compare to one another?
HS.N-Q.1: Use units as a way to understand problems from a variety of contexts (e.g., science, history, and culture), including those of Montana
American Indians, 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.
High School Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. convert to consistent units within a
1. appropriate units
1. that consistent units are important in
problem.
understanding problems, solving multistep problems, using formulas, and in
scaling axes when graphing.
HS.N-Q.2: Define appropriate quantities for the purpose of descriptive modeling.
High School Enduring Understandings
Students will know…
Students will understand…
1. appropriate units
1. that scaling is a good tool to use when
modeling.
Students will be able to…
1. take units that are too large to work with
and scale them down to a model
appropriate for the classroom or units that
too small to work with and scale them up.
HS.N-Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
High School Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. report solutions with the appropriate
1. that the solution is only as accurate as the
1. what is the least accurate measurement
accuracy.
least accurate measurement.
High School – Real Number System
Essential Questions:
1. Why do we use numbers, what are their properties, and how does our number system function?
2. Why do we use estimation and when is it appropriate?
3. What makes a strategy effective and efficient and the solution reasonable?
4. How do numbers relate and compare to one another?
Essential Vocabulary: rational exponent, exponent, base, rational exponent form, radical form
HS.N-RN.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 51/3 to be the cube root of 5 because we want
(51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. rational exponent
1. that the rules apply whether the exponents 1. identify rational exponents.
are integers or rational numbers.
2. apply the rules of exponents using integers
to expressions with rational exponents.
HS.N-RN.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that given an operation sometimes rational 1. rewrite an expression in rational form into
1. rational form
radical form and vice versa.
form is needed and sometimes radical form
2. radical form
is needed.
HS.N-RN.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. perform operations accurately with rational
1. that real numbers are closed with respect
1. rational and irrational number
and/or irrational numbers.
to addition and multiplication.
High School – Vectored and Matrix Quantities
Essential Questions:
1. Why do we use numbers, what are their properties, and how does our number system function?
2. Why do we use estimation and when is it appropriate?
3. What makes a strategy effective and efficient and the solution reasonable?
4. How do numbers relate and compare to one another?
Essential Vocabulary vector, vector quantities, magnitude, direction, directed line segments, components of a vector, terminal point, initial point,
velocity, vector representation of quantities, vector subtraction, parallelogram rule, scalar, scalar multiplication, matrices, matrix multiplication,
square matrices, commutative operation, associative properties, distributive properties, zero matrix, identity matrix, determinant, multiplicative
inverse of a matrix, transformations of vectors, determinant
HS.N-VM.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).
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. vector,
1. that vectors have magnitude and direction.
1. represent vector quantities using
2. vector quantities
appropriate symbols for vectors and
3. magnitude
their magnitudes.
4. direction
5. directed line segments
HS.N-VM.2: (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that to find the components of a vector one 1. accurately find the components of a vector.
1. components of a vector
must subtract the coordinates of an initial
2. terminal point
point from the coordinates of a terminal
3. initial point
point.
HS.N-VM.3: (+) Solve problems from a variety of contexts (e.g., science, history, and culture), including those of Montana American Indians,
involving velocity and other quantities that can be represented by vectors.
Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that problems involving velocity can be
1. solve problems involving velocity and
1. velocity
solved represented by vectors.
other quantities that can be represented by
2. vectors
vectors.
3. vector representation of quantities
HS.N-VM.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
magnitudes 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.
High School Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that the sum or difference of vectors can be 1. add vectors end-to-end, component-wise,
1. vector
found using the parallelogram rule.
and by parallelogram rule.
2. parallelogram rule
2. determine the magnitude and direction of
3. magnitude
the sum of two vectors.
4. direction
3. subtract vectors using additive inverse and
5. vector subtraction
represent vector subtraction graphically.
6. additive inverse
7. vector components
HS.N-VM.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 ‖ ‖ | | . Compute the direction of cv
knowing that when | |
0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. represent scalar multiplication graphically
1. that multiplication of a vector by a scaler
1. vector
and perform scalar multiplication
changes the magnitude of the vector.
2. scalar
component-wise.
2. that the scaler changes the size of the
3. scalar multiplication
2. compute the magnitude of a scalar multiple
vector.
4. magnitude
cv and compute the direction of cv when
5. direction
| |
0.
HS.N-VM.6: (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that matrices can be used to represent and 1. use matrices to represent and manipulate
1. matrices
data
manipulate data.
2. payoffs
3. incidence relationships
4. network
HS.N-VM.7: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. matrices
1. that new matrices can be produced by
1. multiply matrices by scalers.
2. scalars
multiplying matrices by scalers.
HS.N-VM.8: (+) Add, subtract, and multiply matrices of appropriate dimensions.
Enduring Understandings
Students will know…
Students will understand…
1. matrices
1. that operations can be performed on
matrices.
Students will be able to…
1. add, subtract, and multiply matrices of
appropriate dimensions.
HS.N-VM.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.
Enduring Understandings
Students will understand….
Students will be able to…
Students will know…
1. that the commutative property does not
1. apply the associative and distributive
1. matrix multiplication
work for square matrices.
properties in matrix multiplication.
2. square matrices
3. commutative operation
4. associative properties
5. distributive properties.
HS.N-VM.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. apply the zero and identity matrices in
1. that the zero and identity matrices play a
1. zero matrix
addition and multiplication.
similar role in matrix addition and
2. identity matrix
multiplication to 0 and 1 in real numbers.
3. determinant
2. that the determinant of a square matrix is
4. multiplicative inverse of a matrix
nonzero if and only if the matrix has a
multiplicative inverse..
HS.N-VM.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.
HS.N-VM.11
Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that multiplication of a vector by a
1. multiply a vector (column matrix) by a
1. vector
matrix produces another vector.
matrix of suitable dimensions to produce
2. matrix
another vector.
3. transformations
2. work with matrices as transformations of
vectors.
HS.N-VM.12: (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that matrices can be represented as
1. work with 2x2 matrices as transformations
1. matrices
transformations of the plane.
of the plane.
2. transformation
2. interpret the absolute value of the
3. plane
determinant in terms of area.
4. absolute value
5. determinant
6. area