NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
Grade 12 Pre-Calculus Module 2
Vectors and Matrices
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
•
•
•
•
•
Classroom teacher
Math trainer or coach
Principal or school leader
District representative / leader
Other
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
•
•
•
•
Participants will understand the development of the
notion of linearity in an algebraic context (“Which
familiar algebraic functions are linear?”).
Participants understand complex numbers and their
corresponding transformations in the complex plane.
Participants discover matrices and their
transformations.
Participants will enrich their knowledge and experience
in order to implement Module 2 with confidence and
success.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Today
• Module Overview
• Topic A – Networks and Matrices
• Lunch
• Topic B – Linear Transformations in Planes and Space
• Mid-Module Assessment
• Topic C – Systems of Linear Equalities
Tomorrow
• Topic D – Vectors in Plane and Space
• Topic E – First Person Videogames: Projection Matrices
• End-of-Module Assessment
• Discussion for Implementation
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 2: Vectors and Matrices
Module Overview
•
•
•
•
•
•
5 Topics
27 Lessons
35 days
Mid-Module Assessment (After Topic B)
End of Module Assessment (After Topic E)
Number & Quantity: Vector & Matrix Quantity
• N-VM 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
• Algebra: Reasoning with Equations and Inequalities
• A-REI 8, 9
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 2: Vectors and Matrices
Topic A
• Introduces matrices as a tool to organize networks.
• Add, subtract, and multiply square and rectangular matrices.
Topic B
• Properties of matrix operations.
• Linear transformations represented by matrices.
Topic C
• Solve systems of equations using inverse matrices.
Topic D
• Vectors, magnitude and direction.
• Vector applications.
Topic E
• Designing computer games (projecting 3-D images on to 2-D planes).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic A: Networks and Matrices
Topic Overview
• Lessons 1-3
• Introduces students to networks and directed graphs.
• Introduce matrices as a tool to organize networks.
• Scalar multiplication.
• Add, subtract, and multiply square matrices.
• Add, subtract, and multiply rectangular matrices.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
A network diagram is a graph in which the vertices are
represented by circles and the vertices are connected by
segments, edges, or arcs.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
Exploratory Challenge #1
• How many ways can we travel from
City 1 to City 4?
• 3 ways
• How many ways can we reasonably travel
from City 4 to City 1?
• Only 1 way
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
Exploratory Challenge #2
• What might the loop at City 1 represent?
• Tour bus
• How many ways can you travel from City 1
to City 4 if you want to stop in City 2 and
make no other stops?
• 3 routes from City 1 to City 2
• 2 routes from City 2 to City 4
• Therefore 6 routes from City 1 to City 4 through City 2.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
Exploratory Challenge #2
• How many possible ways are there to travel
from City 1 to City 4 without repeating a city?
•
•
•
•
•
•
17 possible ways
City 1 to City 4 with no stops: No routes
City 1 to City 4 with a stop in City 2: 6 routes
City 1 to City 4 with a stop in City 3: 1 route
City 1 to City 4 via City 3 then City 2: 4 routes
City 1 to City 4 via City 2 then City 3: 6 routes
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1: Introduction to Network Diagrams
Exploratory Challenge #2
Cities of Origin
Destination Cities
© 2012 Common Core, Inc. All rights reserved. commoncore.org
1
2
3
4
1
1
3
1
0
2
2
0
2
2
3
2
1
0
1
4
0
2
1
0
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1: Introduction to Network Diagrams
Exploratory Challenge #2
Cities of Origin
Destination Cities
© 2012 Common Core, Inc. All rights reserved. commoncore.org
1
2
3
4
1
11
33
11
00
2
2
2
0
0
2
2
2
2
3
22
11
0
1
4
00
22
1
0
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
Exercises 8 - 14
• What is the value of 𝑟2,3 ? What
does it represent?
•2
• The number of direct routes from City 2
to City 3
1
3
1
0
2
0
2
2
2
1
0
1
0
2
1
0
© 2012 Common Core, Inc. All rights reserved. commoncore.org
• What is the value of 𝑟2,3 ∙ 𝑟3,1 ?
What does it represent?
•4
• The number of 1-stop routes from City
2 to City 1 stopping in City 3
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: Introduction to Network Diagrams
Exercises 8 - 14
• Write an expression for the total
number of one-stop routes from
City 4 to City 1 and determine the
number of routes stopping in one
city.
1
3
1
0
2
0
2
2
2
1
0
1
0
2
1
0
© 2012 Common Core, Inc. All rights reserved. commoncore.org
• 𝑟4,2 ∙ 𝑟2,1 + 𝑟4,3 ∙ 𝑟3,1
•6
• Do you notice a pattern?
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 2 & 3: Networks and Matrix Arithmetic
YOUR TURN!
Lesson 2 Opening Exercise
Lesson 2 Exploratory Challenge
Lesson 3 Opening Exercise
Lesson 3 Exploratory Challenge
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Networks and Matrix Arithmetic
Lesson Summary
MATRIX SCALAR MULTIPLICATION. Let 𝑘 be a real number and let 𝐴 be an 𝑚 × 𝑛 matrix
whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖,𝑗 . Then the scalar product 𝑘 ⋅ 𝐴 is the 𝑚 × 𝑛
matrix whose entry in row 𝑖 and column 𝑗 is 𝑘 ⋅ 𝑎𝑖,𝑗 .
MATRIX SUM. Let 𝐴 be an 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖,𝑗 and
let 𝐵 be an 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑏𝑖,𝑗 . Then the matrix sum
𝐴 + 𝐵 is the 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖,𝑗 + 𝑏𝑖,𝑗 .
MATRIX DIFFERENCE. Let 𝐴 be an 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is
𝑎𝑖,𝑗 and let 𝐵 be an 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑏𝑖,𝑗 . Then the
matrix difference 𝐴 − 𝐵 is the 𝑚 × 𝑛 matrix whose entry in row 𝑖 and column 𝑗 is 𝑎𝑖,𝑗 −
𝑏𝑖,𝑗 .
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3: Matrix Arithmetic in its Own Right
MATRIX PRODUCT. Let 𝑨 be an 𝒎 × 𝒏 matrix whose entry in row 𝒊 and column 𝒋 is 𝒂 and let 𝑩 be an 𝒏 × 𝒑 matrix
𝒊𝒋
whose entry in row 𝒊 and column 𝒋 is 𝒃 . Then the matrix product 𝑨𝑩 is the 𝒎 × 𝒑 matrix whose entry in row 𝒊 and
𝒊𝒋
column 𝒋 is 𝒂 𝒃 + 𝒂 𝒃 + ⋯ + 𝒂 𝒃 .
𝒊𝟏 𝟏𝒋
𝒊𝟐 𝟐𝒋
𝒊𝒏 𝒏𝒋
,
,
,
,
,
,
IDENTITY MATRIX. The 𝒏 × 𝒏 identity matrix is the matrix whose entry in row 𝒊 and column 𝒊 for 𝟏 ≤ 𝒊 ≤ 𝒏 is 1, and
whose entries in row 𝒊 and column 𝒋 for 𝟏 ≤ 𝒊, 𝒋 ≤ 𝒏 and 𝒊 ≠ 𝒋 are all zero. The identity matrix is denoted by 𝑰. The 𝟐
𝟏
𝟏
𝟎
and the 𝟑 × 𝟑 identity matrix is
× 𝟐 identity matrix is
𝟎
𝟎
𝟏
𝟎
explicitly stated, then the size is implied by context.
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
. If the size of the identity matrix is not
ZERO MATRIX. The 𝒎 × 𝒏 zero matrix is the 𝒎 × 𝒏 matrix in which all entries are equal to zero. For example, the 𝟐 × 𝟐
𝟎
𝟎 and the 𝟑 𝟑 zero matrix is
×
𝟎
𝟎 𝟎
𝟎
then the size is implied by context.
zero matrix is 𝟎
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
. If the size of the zero matrix is not specified explicitly,
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing Up Topic A
Let’s Review
What major concepts and/or themes did you notice in Topic A?
What connections do you see too Module 1?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic B: Linear Transformations of Planes
and Space
Topic Overview
•
•
•
•
•
10 lessons (4 - 13)
Transformations of points in 2-D and 3-D.
Composition of transformations.
Properties of matrix addition and multiplication.
Another use for matrices - encryption.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: Linear Transformations Review
What are the conditions for linearity?
• 𝐿 𝑥 + 𝑦 = 𝐿 𝑥 + 𝐿(𝑦)
• 𝐿 𝑎∙𝑥 = 𝑎∙𝐿 𝑥
What is the form of the matrix is used to model complex number
multiplication?
𝑎 −𝑏
𝑏 𝑎
A point in 2-dimensions (ℝ2 ) can be represented by what matrix?
𝑥1
•
𝑥2
A point in 3-dimensions (ℝ3 ) can be represented by what matrix?
•
•
𝑥1
𝑥2
𝑥3
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 4: Linear Transformations Review
Problem Set #5
𝟎 −𝟏
to a complex number 𝒛 will produce a
𝟏 𝟎
𝝅
𝟐 𝟎
pure radians counterclockwise rotation and by multiplying
will produce a pure
𝟐
𝟎 𝟐
dilation with a factor of 2. So, he thinks he can add these two matrices, which will produce
𝝅
𝟐 −𝟏
(
) that will rotate 𝒛 by radians counterclockwise and dilate 𝒛 with a factor of 2. Is
𝟐
𝟏 𝟐
he correct? Explain your reason.
Wesley noticed that by multiplying the matrix
No, he is not correct. For the general transformation of complex numbers, the form is
𝒂 −𝒃 𝒙
𝑳 𝒁 =
𝒚 .
𝒃 𝒂
𝒃
𝒂 −𝒃
By multiplying the matrix
to 𝒛, it rotates 𝒛 an angle 𝒂𝒓𝒄𝒕𝒂𝒏 𝒂 , and
𝒃 𝒂
dilates 𝒛 with a factor of 𝒂𝟐 + 𝒃𝟐 .
𝟏
𝒂𝒓𝒄𝒕𝒂𝒏
= 𝟐𝟔. 𝟓𝟔𝟓° ,
𝟐
of a factor of 2.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝟏
𝟐
+ (𝟐)𝟐 = 𝟓 , which are not
𝝅
𝟐
radians or a dilation
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Coordinates of Points in Space
𝟐
𝟓
,𝐲 =
, compute 𝐳 = 𝐱 + 𝐲 and draw the associated
𝟑
𝟏
parallelogram.
Let 𝐱 =
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: Coordinates of Points in Space
𝟑
𝟏
Let 𝐱 = 𝟏 and 𝐲 = 𝟑 . Compute 𝐳 = 𝐱 + 𝐲, and then plot of these three
𝟏
𝟏
points.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Linear Transformations as Matrices
• Matrices represent linear transformations in ℝ2 → ℝ2 and also ℝ3 → ℝ3 .
• The identity matrix functions like the number 1 in the real number system.
• The zero matrix functions like the number 0 in the real number system.
• Vector addition is commutative, associative, and distributive.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Linear Transformations applied to Cubes
• We will use GeoGebra to explore linear transformations on cubes.
• Let’s do Exploratory Challenges 1 & 2 together!
GeoGebra Transforming Cubes
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Linear Transformations applied to Cubes
Lesson Summary
For a matrix 𝑨, the transformation 𝑳
𝒙
𝒚
𝒛
𝒙
= 𝑨 ⋅ 𝒚 is a function from points in space to
𝒛
points in space.
Different matrices induce transformations such as rotation, dilation, and reflection.
𝒂 𝟎 𝟎
The transformation induced by a diagonal matrix 𝑨 = 𝟎 𝒃 𝟎 will scale by 𝒂 in the
𝟎 𝟎 𝒄
direction parallel to the 𝒙-axis, by 𝒃 in the direction parallel to the 𝒚-axis and by 𝒄 in the
direction parallel to the 𝒛-axis.
𝟏
𝟎
𝟎
𝐜𝐨𝐬 𝜽 𝟎 −𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
The matrices 𝟎 𝐜𝐨𝐬 𝜽 −𝐬𝐢𝐧 𝜽 ,
𝟎
𝟏
𝟎
, and 𝐬𝐢𝐧 𝜽
𝟎 𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝐬𝐢𝐧 𝜽 𝟎 𝐜𝐨𝐬 𝜽
𝟎
induce rotation by 𝜽 about the 𝒙, 𝒚 and 𝒛 axes, respectively.
.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
−𝐬𝐢𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝟎
𝟎
𝟎
𝟏
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Describe the geometric effect of the transformation 𝑳𝑩
𝒙
𝒚
𝒙
=𝑩⋅ 𝒚 .
The transformation produced by matrix 𝐵 has the
effect of dilation from the origin with scale factor 4.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Describe the geometric effect of the transformation 𝑳𝑨
The transformation produced by matrix
𝐴 has the effect of rotation by 45°.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝒙
𝒚
𝒙
=𝑨⋅ 𝒚 .
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Draw the image of the unit square under the transformation 𝑳𝑩
The large green square is the
image of the original unit
square under the
transformation produced by 𝐵 .
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝒙
𝒚
𝒙
=𝑩⋅ 𝒚
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Draw the image of the transformed square under the transformation 𝑳𝑨
𝒙
𝒚.
The tilted red square is the image of
the green square under the
transformation produced by 𝑨.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝒙
𝒚
=𝑨⋅
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Describe the geometric effect on the unit square of performing first 𝑳𝑩 then 𝑳𝑨 .
If we dilate with factor 4 then rotate by
45° then the net effect is a rotation by
45° and dilation with scale factor 4.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Compute the matrix product 𝑨𝑩.
The matrix product is 𝐴𝐵 = 2 2
2 2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
−2 2 .
2 2
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 8 & 9: Composition of Linear
Transformations
𝑨=
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−
,𝑩=
𝟒 𝟎
𝟎 𝟒
For each pair of matrices 𝑨 and 𝑩 given:
• Describe the geometric effect of the transformation 𝑳𝑨𝑩
𝒙
𝒚
𝒙
= 𝑨𝑩 ⋅ 𝒚 .
The transformation produced by matrix 𝐴𝐵 has the effect
of rotation by 45° while scaling by 4.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 8 & 9: Composition of Linear
Transformations
YOUR TURN!
• Lesson 9 Opening Exercises
• Lesson 9 Exploratory Challenge #1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10: Matrix Multiplication if NOT
Commutative
Exercise 1
Let 𝑨 equal the matrix that corresponds to a 𝟒𝟓° rotation counterclockwise
and 𝑩 equal the matrix that corresponds to a reflection across the 𝒚-axis.
Verify that matrix multiplication does not commute by finding the
products 𝑨𝑩 and 𝑩𝑨.
𝑨𝑩 =
𝑩𝑨 =
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝟐
𝟐
−
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
−𝟏
𝟎
𝟎
𝟏
−𝟏
𝟎
𝟎
=
𝟏
𝟐
𝟐
−
𝟐
𝟐 =
𝟐
𝟐
𝟐
𝟐
𝑨𝑩 ≠ 𝑩𝑨
𝟐
𝟐
−
−
𝟐
𝟐
𝟐
𝟐
−
𝟐
𝟐
𝟐
−
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
𝟐
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: Matrix Multiplication if NOT
Commutative
YOUR TURN!
Lesson 10 Exercise 4
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 10: Matrix Multiplication if NOT
Commutative
What did you discover about the matrices?
• 𝐴𝐵 = 𝐵𝐴
Does this mean matrix multiplication is commutative? Explain.
• No. This is a special case because the matrices are in the form
𝑎
𝑏
−𝑏
.
𝑎
What is the relationship between these matrices and complex numbers?
• Matrices in this form can be used to represent a corresponding complex number.
Multiplying these matrices is the same as multiplying two complex numbers.
Is the multiplication of two complex numbers commutative?
𝑎 −𝑏
have the same
𝑏 𝑎
product, but this does not mean that matrix multiplication is commutative.
• Yes. Therefore, the product of two matrices in the form
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 11: Matrix Addition is Commutative
Lesson 12: Matrix Multiplication is Distributive and
Associative
YOUR TURN!
Lesson 12 Example 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 13: Using Matrix Operations for Encryption
YOUR TURN!
Lesson 13 Example 1 and Exercise 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing Up Topic B
Let’s Review
What major concepts and/or themes did you notice in Topic B?
Have the topics and themes of Topics A and B helped you (and students)
understand matrices and their operations and transformations?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid-Module Assessment
• Do the problems in the Mid-Module
Assessment
• As you work, think about the following:
• Which lesson(s) does this assessment item tie to?
• Is there vocabulary that students may struggle
with?
• Can this item be used as part of a quiz for Topic A
or B?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic C: Systems of Linear Equations
Topic Overview
• 3 lessons (14 – 16).
• Matrices are used to solve systems of linear
• If 𝐿𝑥 = 𝑏, then 𝑥 = 𝐿−1 𝑥
• Real world applications
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lessons 14 & 15: Solving Equations Involving
Linear Transformations of the Coordinate Plane
Given
2𝑥 + 5𝑦 = 4
3𝑥 − 8𝑦 = −25
𝑥
4
a) Write this as a linear transformation if 𝑥 = 𝑦 and 𝑏 =
.
−25
•
𝟐
𝟑
𝟒
𝟓 𝒙
=
−𝟐𝟓
−𝟖 𝒚
b) Does L have an inverse? If it does, compute L−1 b and verify that the
coordinates are solutions to the system of equations.
−𝟖
−𝟑
−𝟖 −𝟓
−𝟑 𝟐
𝟏
• 𝑳−𝟏 = 𝟐(−𝟖)−(𝟓)(𝟑)
𝟏
• 𝑳−𝟏 𝒃 = −𝟑𝟏
© 2012 Common Core, Inc. All rights reserved. commoncore.org
−𝟓
𝟐
𝟏
𝟒
𝟗𝟑
= −𝟑𝟏
−𝟐𝟓
−𝟔𝟐
=
−𝟑
𝟐
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 14, 15 & 16: Solving Equations Involving
Linear Transformations of the Coordinate Plane
YOUR TURN!
Lesson 15 Exercise 2
Lesson 16 Exercise 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing up Topic C
Let’s Review
What major concepts and/or themes did you notice in Topic C?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic D: Vectors in Plane and Space
Topic Overview
•
•
•
•
•
•
8 lessons (17 – 24)
Formally define vector
Vector transformations in 2-D and 3-D
Vector addition, subtraction and scalar multiplication
Vector magnitude and direction
Parametric equations tie work with functions to linearity to
vectors
• Real world applications
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Vectors in the Coordinate Plane
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Vectors in the Coordinate Plane
After the first earthquake shifted point 5 feet east and 10 feet north, a second
earthquake hit a town and shifted all point 6 feet east and 9 feet south.
Draw vector (v) to represent the first
earthquake.
Draw vector (t) to represent the shift of
the second earthquake.
Which earthquake shifted all point in the
town further?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Vectors in the Coordinate Plane
The magnitude of a vector v = <a, b> is the length of the line segment from the
origin to the point (a, b) in the coordinate plane, which we denote 𝐯
How can we find the magnitude of vector
v = <a, b>?
Write the general formula for the magnitude
of a vector.
v = 𝑎2 + 𝑏 2
What is the magnitude of vectors v and t?
v = 52 + 102 = 125
t = 6 2 + −9 2 = 117
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Vectors in the Coordinate Plane
What is the total shift after both earthquakes?
v+t=?
<5, 10> + <6, -9>
<5 + 6, 10 + (-9)>
<11, 1>
© 2012 Common Core, Inc. All rights reserved. commoncore.org
v+t
(11, 1)
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 18: Vectors and Translation Maps
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 18: Vectors and
Translation
Maps
𝟑
What is the magnitude of a vector in ℝ ?
𝑥2 + 𝑦2 = 𝑤2
𝑤2 + 𝑧2 = 𝑣2
𝑣2 − 𝑧2 = 𝑤2
𝑥2 + 𝑦2 = 𝑣2 − 𝑧2
𝑥2 + 𝑦2 + 𝑧2 = 𝑣2
𝑣 =
𝑥2 + 𝑦2 + 𝑧2
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: Directed Line Segments and Vectors
Several vectors, represented by arrows, are shown below. For each
vector, state the initial point, terminal point, component form of the
vector and magnitude
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: Directed Line Segments and Vectors
Let 𝐮 =
and 𝐯 = 𝟒 𝟑 .
𝟐
𝟓
− ,
,
Draw a diagram to illustrate 𝐯 and 𝐮 and then find 𝐯 + 𝐮 using the
parallelogram rule.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: Directed Line Segments and Vectors
Let 𝐮 =
and 𝐯 = 𝟒 𝟑 .
𝟐
𝟓
− ,
,
Draw a diagram to illustrate 𝟐𝒗 and then find 𝟐𝒗 + 𝒖 using the
parallelogram rule.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: Directed Line Segments and Vectors
Let 𝐮 =
and 𝐯 = 𝟒 𝟑 .
𝟐
𝟓
− ,
,
Draw a diagram to illustrate – 𝐯 and then find 𝐮 − 𝐯 using the
parallelogram rule.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 20: Vectors and Stone Bridges
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 20: Vectors and Stone Bridges
YOUR TURN!
Let’s do the Lesson 20
Exploratory Challenge.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: Vectors and the Equation of a Line
Lesson Summary
Lines in the plane and lines in space can be described by either a vector equation or a set of parametric equations.
𝒂
𝒃
. If the slope of line 𝓵 is defined, then 𝒎 = .
𝒂
𝒃
Let 𝓵 be a line in the plane that contains point 𝒙𝟏 , 𝒚𝟏 and has direction vector 𝒗 =
A vector form of the equation that represents line 𝓵 is
𝒙
𝒙𝟏
𝒂
𝒚 = 𝒚𝟏 + 𝒃 𝒕.
Parametric equations that represent line 𝓵 are
𝒙 𝒕 = 𝒙𝟏 + 𝒂𝒕
𝒚 𝒕 = 𝒚𝟏 + 𝒃𝒕.
𝒂
Let 𝓵 be a line in space that contains point (𝒙𝟏 , 𝒚𝟏 , 𝒛𝟏 ) and has direction vector 𝒗 = 𝒃 .
𝒄
A vector form of the equation that represents line 𝓵 is
𝒙𝟏
𝒂
𝒙
𝒚 = 𝒚𝟏 + 𝒃 𝒕.
𝒛𝟏
𝒛
𝒄
Parametric equations that represent line 𝓵 are
𝒙 𝒕 = 𝒙𝟏 + 𝒂𝒕
𝒚 𝒕 = 𝒚𝟏 + 𝒃𝒕
𝒛 𝒕 = 𝒛𝟏 + 𝒄𝒕.
.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: Vectors and the Equation of a Line
Lesson 22: Linear Transformations of Lines
Let’s do Exercises 1 – 3 together!
YOUR TURN!
Lesson 21 Exercises 4 & 6
Lesson 22 Exercise 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 23 & 24: Why are Vectors Useful?
Vectors in the NFL
Which of these phenomena can be represented as a vector?
• Position of a moving object
• Wind
• Position of a ball that has been thrown
• Temperature
• Mass
• Velocity of a ball that has been thrown
• Volume
• Water current
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 23 & 24: Why are Vectors Useful?
Let’s do Lesson 23 Example 1 together!
YOUR TURN!
Lesson 23 Exercise 1
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing up Topic D
Let’s Review
What major concepts and/or themes did you notice in Topic D?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic E: First-person Videogames: Projection
Matrices
Topic Overview
•
•
•
•
3 lessons (25 - 27)
Project 3-D objects onto a 2-D plane
Explore ALICE
Design own video game
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: First-Person Computer Games
ALICE 3.1
Vanishing Points
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 25: First-Person Computer Games
Let’s try the Exercise 1 together!
In this drawing task, the “eye” or
the “camera” is the point, and
the shaded figure is the “TV screen”.
The cube is in the 3-D universe of the
Computer game.
Using lines drawn from each vertex of the
cube to the point, draw the image of the
3-D cube on the “TV screen”.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 26: Projecting a 3-D Object onto a
2-D Plane
Project the point (-1, 4, 5) onto the plane y = 1.
• We need the value of 𝒕 that makes 𝟒𝒕 = 𝟏
𝟏
𝟒
𝟏
𝟒
𝟓
𝟒
• so 𝒕 = . Thus the image is (− , 𝟏, ).
Project the point (9, 5, -8) onto the plane z = 3.
• We need the value of 𝒕 that makes −𝟖𝒕 = 𝟑
𝟑
𝟖
• so 𝒕 = − . Thus the image is (−
© 2012 Common Core, Inc. All rights reserved. commoncore.org
𝟐𝟕
𝟏𝟓
, − , 𝟑).
𝟖
𝟖
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 26: Projecting a 3-D Object onto a
2-D Plane
Let’s do the Example (Rotations in 3-D) together!
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 27: Designing Your Own Game
Students will use ALICE 3.1 to design their own video game.
Specific directions are included.
Let’s look at Lesson 27 together!
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Summing up Topic E
Let’s Review
What major concepts and/or themes did you notice in Topic E?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
End-of-Module Assessment
• Do the problems in the End-of-Module
Assessment
• As you work, think about the following:
• Which lesson(s) does this assessment item tie to?
• Is there vocabulary that students may struggle
with?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Themes of Module 2: Vectors and
Matrices
•
•
•
•
•
•
•
Matrices as tools.
Matrix operations and properties.
Solving systems of equations using inverse matrices.
Linear transformations represented by matrices and vectors.
Vectors, magnitude, and direction.
Vector applications.
Designing computer games (projecting 3-D images on to 2-D
planes).
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Biggest Takeaway
•
What is your biggest takeaway with respect to Module 2?
•
How can you support successful implementation at your
school/s given your role?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
