Content Area
Standard
Math
4.HS High School
Strand
N-VM. Number and Quantities – Vector and Matrix Quantities
Content Statement
CPI#
Cumulative Progress Indicator (CPI)
ACSSSD
Objectives
Represent and model with vector quantities.
4.HS.NVM.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).
1. Distinguish between vector
quantities, which have both
magnitude and direction; and
scalar quantities, which have
magnitude only.
2. Recognize the magnitude of a
vector as its length.
3. Recognize the direction of a
vector as an angle.
4. Demonstrate an understanding
that any vector can be considered
a sum of two or more vectors and
can be separated into component
vectors.
5. Demonstrate the ability to
separate vectors into component
vectors.
6. Recognize that component
vectors run parallel to the
coordinate axes; one in the
direction of x and one in the
direction of y.
7. Identify the direction of vectors
considering all 360° of motion.
8. Draw a vector in a coordinate
plane and describe its direction.
9. Demonstrate an understanding of
and identify symbols for vectors
and their magnitudes (v-
vector,|v|-magnitude, ||v||magnitude, v).
4.HS.NVM.2
Find the components of a vector by
subtracting the coordinates of an initial
point from the coordinates of a terminal
point.
4.HS.NVM.4
Add and subtract vectors.
1. Demonstrate an understanding
that subtracting vectors is the
same as adding an opposite
vector or negative of the second
vector to the first vector {a-b=
a+(-b)},
2. Demonstrate an understanding
that to subtract a vector, the
vector to be subtracted must first
be reversed {(6,4)-(3,2)= (6,4) +
(-3,-2)}.
3. Demonstrate an understanding
that the coordinates must be
subtracted in the proper order
(6-3,4-2) = (3,2).
Perform operations on vectors.
a.
Add vectors end-to-end, component-wise, 1. Demonstrate an understanding
and by the parallelogram rule. Understand
that adding vectors end-to-end
that the magnitude of a sum of two
involves joining vectors
vectors is typically not the sum of the
sequentially so that the tail of
magnitudes.
each vector joins the head of the
preceding vector to begin
forming a polygon; and that the
tail of the first vector is joined to
the head of the last vector to
form the resultant vector, which
is the sum of all the vectors.
2. Demonstrate an understanding
that adding vectors componentwise involves adding the
components in the proper order
(6,4)+(3,2)={(6+3),(4+2)}=(9,6)
b.
Given two vectors in magnitude and
direction form, determine the magnitude
and direction of their sum.
3. Demonstrate an understanding
that adding vectors using the
parallelogram rule involves
placing vectors tail to tail and
then completing the
parallelogram by placing a copy
of the second vector at the head
of the first vector and a copy of
the first vector at the head of the
second vector; the resultant
vector or sum of the vectors is
found by drawing a vector
diagonally from where the two
tails meet to where the two heads
meet.
4. Demonstrate an understanding
that the magnitude of a sum of
two vectors is typically not the
sum of the magnitudes: the
resultant vector or sum of the
vectors is determined by both
magnitude and direction.
1. Demonstrate an understanding
that adding vectors componentwise involves drawing the
vectors in the coordinate plane,
finding the x- and y- components
of the vectors (magnitude or
length and direction or angle) by
forming a right triangle from
each vector with the legs
representing the x- and ycomponents and the original
vector representing the
hypotenuse, using standard
triangle trigonometry (sine and
cosine functions) to solve for
each x- and y- component vector,
adding the x-component from the
first vector and the x-component
from the second vector to find
the sum for the x-component,
adding the y-component from the
first vector to the y-component of
the second vector to find the sum
for the y-component, using the xand y- components of the sum as
the legs of a right triangle with
the resultant vector as the
hypotenuse, using the
Pythagorean Theorem to find the
length of the hypotenuse which is
the magnitude of the resultant
vector and using the inverse
tangent function to find angle
which is the direction of the
resultant angle.
2. Demonstrate an understanding
that adding vectors using the
parallelogram rule involves
placing vectors tail to tail and
then completing the
parallelogram by placing a copy
of the second vector at the head
of the first vector and a copy of
the first vector at the head of the
second vector; the resultant
vector or sum of the vectors is
found by drawing a vector
diagonally from where the two
tails meet to where the two heads
meet; the magnitude and
direction are found using
standard triangle trigonometry to
find hypotenuse (magnitude) and
3.
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.
1.
2.
3.
4.
angle (direction) of the resultant
vector.
Demonstrate an understanding of
triangle trigonometry (sine ratio,
cosine ratio, tan ratio, inverse
tan) to solve for the angle or
direction of the vector and
Pythagorean theorem: c2=a2+b2
to solve for the hypotenuse or
magnitude of the vector.
Demonstrate an understanding
that subtracting vectors is the
same as adding an opposite
vector or negative of the second
vector to the first vector (a-b=
a+(-b).
Demonstrate an understanding
that subtracting vectors involves
joining vectors so that the tail of
the opposite or negative of the
second vector joins the head of
the first vector to begin forming a
polygon; and that the tail of the
first vector is joined to the head
of the second vector to form the
resultant vector, which is the
difference of the vectors.
Demonstrate an understanding
that vectors can be subtracted by
joining the vectors tail to tail and
the resultant vector is drawn
between the heads of each vector.
Demonstrate an understanding
that subtracting vectors using the
parallelogram rule involves
placing vectors tail to tail to
begin forming a parallelogram
and then completing the
parallelogram by placing a copy
of the first vector at the head of
the second vector and a copy of
the second vector at the head of
the first vector; the resultant
vector or sum of the vectors is
found by drawing a vector
diagonally from where the tail of
the copy of the first vector meets
the tail of the inverse of the
second vector (drawn in the
reverse direction of the second
vector); the magnitude and
direction are found using
standard triangle trigonometry to
find hypotenuse(magnitude) and
angle (direction) of the resultant
vector.
5. Demonstrate an understanding
that subtracting vectors
component-wise involves the
same method as adding vectors
component-wise except the
negative or inverse of the second
vector is used.
6. Demonstrate an understanding of
triangle trigonometry (sine ratio,
cosine ratio, tan ratio, inverse
tan) to solve for the angle or
direction of the vector and
Pythagorean theorem: c2=a2+b2
to solve for the hypotenuse or
magnitude of the vector.
Perform operations on matrices and use matrices in applications.
4.HS.NVM.6
Use matrices to represent and manipulate
data, e.g., to represent payoffs or
incidence relationships in a network.
4.HS.NVM.7
Multiply matrices by scalars to produce
new matrices, e.g., as when all of the
payoffs in a game are doubled.
4.HS.NVM.8
Add, subtract, and multiply matrices of
appropriate dimensions.
1. Identify the dimensions of a
matrix as the number of rows and
columns it has (2 rows and 2
columns, dimensions are 2X2).
2. Identify each number in the
matrix as an entry.
3. Identify each entry in the matrix
by its address which is
determined by the row and
column where the entry is
located.
4. Demonstrate the ability to
interpret data in a matrix.
1. Demonstrate an understanding
that scalar multiplication
involves multiplying a matrix by
a constant or real number.
2. Demonstrate an understanding
that multiplying a matrix by a
constant involves multiplying
each entry by the constant given.
1. Identify two matrices as being
equal; having the same
dimensions and equivalent
corresponding entries.
2. Demonstrate an understanding
that that adding matrices of
appropriate dimensions involves
adding each of the corresponding
entries.
3. Demonstrate an understanding
that that subtracting matrices of
appropriate dimensions involves
subtracting each of the
corresponding entries.
4. Demonstrate an understanding
that that adding and subtracting
more than two matrices involves
adding and/or subtracting the
matrices in the proper order. For
example, to solve A+B-C first
add each of the corresponding
entries for A+B and then
subtract each of the
corresponding entries for C,
likewise to solve A-B+C subtract
each of the corresponding entries
for A-B then add the
corresponding entries for C.
5. Demonstrate an understanding
that that multiplication of
matrices of the same dimensions
results in a matrix of those same
dimensions.
6. Demonstrate the ability to
multiply 2X2 matrices.
7. Demonstrate an understanding
that multiplication of 2 matrices
of the same dimensions (2X2)
involves multiplying the first
entry of row 1 of first matrix by
first entry of column 1 of second
matrix and then multiplying
second entry of row I of first
matrix by second entry of
column 1 of the second matrix
and adding those two products
together to find entry for row 1,
column 1 of new matrix. Next,
the firsr entry of row 2 of the first
matrix is multiplied by first entry
of column1 of the second matrix
and then the second entry of row
2 of the first matrix is multiplied
by the second entry of column 1
of the second matrix and the two
products are added together to
find the entry for row 2 column 1
of the new matrix. Next, the first
entry of row 1 of first matrix is
multiplied by first entry of
column 2 of second matrix and
second entry of row 1 first matrix
multiplied by second entry of
column 2 of the second matrix
and the two products are added
together to find entry for row 1
column 2. Next the first entry of
row 2 of first matrix is multiplied
by first entry of column 2 of
second matrix and second entry
of row 2 of first matrix is
multiplied by second entry
column 2 of second matrix and
the two products are added
together to find entry for row 2
column 2 of new matrix.