Grid Morphing Guide Worksheet Grid Morphing Guide
UNIT: LINEAR TRANSFORMATIONS
Lesson 1: Matrices as Functions
Name:
Date:
THE BIG IDEA
A 2x2 matrix isn't just a box of numbers—it's a machine that takes a vector input \(\vec{v}\) and spits out a transformed vector output \(T(\vec{v})\). To see what the whole machine does, we just need to watch where it sends the two "basis vectors": \(\hat{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(\hat{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).
Part 1: The Transformation Map
For the matrix \(M = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}\), calculate where the basis vectors land.
Transform \(\hat{i}\):
\(\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} \dots \\ \dots \end{bmatrix}\)
Show work here
Transform \(\hat{j}\):
\(\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} \dots \\ \dots \end{bmatrix}\)
Show work here
Visualizing the Morph
Original Grid (1x1 Square)
\(\hat{i}\)
\(\hat{j}\)
Apply Matrix
Transformed Grid
Plot your new \(\hat{i}\) and \(\hat{j}\) and connect the vertices.
The Inverse Problem: Building the Machine
Often, we know what we want the grid to look like, and we need to find the matrix that creates that look. Crucial Insight: The first column of the matrix is where \(\hat{i}\) lands. The second column is where \(\hat{j}\) lands.
Challenge: The Diagonal Stretch
Find the matrix that stretches the grid by a factor of 3 horizontally and 0.5 vertically.
1. Identify target locations:
New \(\hat{i}\) should be at:
New \(\hat{j}\) should be at:
2. Construct the Matrix:
[
]
Observation 1
If the matrix is \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\), describe in words what happens to the unit square. (Hint: look at \(\hat{i}\) and \(\hat{j}\) individually first).
Observation 2
What happens to the area of the unit square if we use the matrix \(\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}\)? Why?
The "Collapsing" Matrix
Predict what a 2D grid would look like if both \(\hat{i}\) and \(\hat{j}\) land on the same line (e.g., \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\)). Is it still a "plane"?
Matrix Machine Slides Linear Transformations
THE MATRIX MACHINE
Visualizing matrices as functions that move vectors through geometric space.
Explore the Grid
Map the Basis
The "Machine" Analogy
Level 01
INPUT
Any vector \(\vec{v} = \begin{bmatrix} x \\ y \end{bmatrix}\) in the 2D plane.
FUNCTION
The matrix \(M\). It processes the input using multiplication.
OUTPUT
The transformed vector \(M\vec{v}\).
\[ \vec{v}_{new} = M \vec{v}_{old} \]
"The matrix acts as a rule for moving every single point on the grid."
The "Secret" Cheat Code
Why we only need to track the two unit vectors.
Linear Combinations
Every vector in 2D is just some amount of \(\hat{i}\) and some amount of \(\hat{j}\) .
\(\vec{v} = x \cdot \hat{i} + y \cdot \hat{j}\)
Where they land...
...tells you exactly how the matrix behaves.
\(\hat{i}\)
Goes to the 1st Column
\(\hat{j}\)
Goes to the 2nd Column
Mental Grid Morph
Can you predict the result?
\( M = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \)
1
\(\hat{i}\) lands at (0, 1)
2
\(\hat{j}\) lands at (-1, 0)
Geometric Result
90° Counter-Clockwise Rotation
Matrix Machine Teacher Guide The Matrix Machine
TEACHER FACILITATION GUIDE
DURATION: 55 MIN LEVEL: GRADE 11 MATH
Lesson Objective
Students will transition from algebraic matrix-vector multiplication to a geometric interpretation. By the end of this lesson, students should be able to predict the transformation of a 2D grid by calculating only where the basis vectors land.
The "Hook" Simulation
Before opening the worksheet, use a dynamic geometry tool (GeoGebra or Desmos Matrix Calculator).
Show a unit square on a grid.
Input matrix \(\begin{bmatrix} a & c \\ b & d \end{bmatrix}\).
Drag the slider for 'a' (top left). Watch the square stretch horizontally.
Drag the slider for 'c' (top right). Watch the square "shear" horizontally.
Discussion: "Why does changing this specific number move the grid in this specific way?"
KEY TERMS
Linear Transformation: A mapping between vector spaces that preserves vector addition and scalar multiplication.Basis Vectors: The "building block" vectors \((1,0)\) and \((0,1)\).Span: The set of all possible linear combinations of a set of vectors.
15m
Guided Discovery (Slides 1-3)
Introduce the "Matrix Machine" concept. Focus heavily on Slide 3. Many students miss that matrix columns ARE the images of basis vectors. Have students physically point to the columns and then to the graph.
20m
Worksheet Part 1: Mapping
Students work through Part 1 of the Grid Morphing Guide . Circulate and check for arithmetic errors in matrix-vector multiplication. Ensure they are drawing vectors as arrows starting from the origin.
Check for Understanding: If a student gets \(\begin{bmatrix} 2 \\ 0 \end{bmatrix}\) for \(\hat{i}\), ask them: "Does this look like a stretch or a rotation?"
15m
The Inverse Challenge
Transition to Part 2. This is harder because it's synthesis. Students must work backward from a visual goal to a numeric matrix. Discuss the "Diagonal Stretch" as a class.
05m
Closing & Prediction
Review the "Collapsing Matrix" concept. This seeds the idea of determinants and singular matrices (though those terms aren't required yet).
Differentiation Strategies
Scaffolding (For Support)
Provide a "Column Tracker" sticky note where students can write: "Column 1 = Where \((1,0)\) goes". Encourage color-coding: red pen for \(\hat{i}\) calculations and blue pen for \(\hat{j}\).
Mirrors and Morphs Slides Lesson 2
Mirrors & Morphs
Building a library of standard geometric matrices: reflections, dilations, and shears.
The "Do Nothing" Matrix
IDENTITY
If we want \(\hat{i}\) to stay at \((1,0)\) and \(\hat{j}\) to stay at \((0,1)\)...
\( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)
This is the starting point for every transformation.
Dilation (Scaling)
Horizontal Stretch/Squish
Scale factor \(k\) on the x-axis:
\(\begin{bmatrix} k & 0 \\ 0 & 1 \end{bmatrix}\)
Uniform Scaling
Scale everything by factor \(k\):
\(\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}\)
SCALE FACTOR: 2.0
The Mirror Matrix
Flip Over X-Axis
\(\hat{i}\) stays, \(\hat{j}\) flips.
\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
Flip Over Y-Axis
\(\hat{i}\) flips, \(\hat{j}\) stays.
\(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
Flip Over \(y = x\)
\(\hat{i}\) and \(\hat{j}\) swap places.
\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
The Shear
Think of a deck of cards. You keep the bottom fixed and slide the top.
Horizontal
\(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\)
Vertical
\(\begin{bmatrix} 1 & 0 \\ k & 1 \end{bmatrix}\)
SHEAR
Mirror Maze Activity Mirror Maze Challenge
Mission: Matrix Reflection
Team Name:
Date:
Target Briefing
You are controlling a high-energy laser vector \(\vec{v}\). To hit the target, you must reflect your laser using the correct transformation matrix. Be careful: using the wrong matrix will send the laser into the "Dark Zone" and end the mission!
SECTOR 01
Target: The Y-Axis Mirror
Current Laser: \(\vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}\)
\(\vec{v}\)
Select Mirror Matrix:
\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) \(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
Resulting Vector:
Calculate \(M\vec{v}\)
SECTOR 02
Target: The "Diagonal" Deflector
Reflect across the line \(y = x\). Current Laser: \(\vec{v} = \begin{bmatrix} 1 \\ 5 \end{bmatrix}\)
Construct Mirror Matrix:
[
]
New Coordinates:
(\(x'\), \(y'\))
Level 3: The Super-Morph
Dual Matrix Maneuver
To hit the final target at \((6, 0)\), you must first stretch the laser \(\vec{v} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) by a factor of 3, then shear it horizontally by a factor of 2.
Final Matrix Product
The Sweep Slides Lesson 3
THE SWEEP
How do we rotate any object by any angle? Bringing Trigonometry into the Matrix.
"Rotating the grid is just rotating the basis vectors."
Step 1: Rotate \(\hat{i}\)
DERIVATION
Original \(\hat{i} = (1, 0)\).
The First Column
\(\begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix}\)
(\(\cos \theta\), \(\sin \theta\))
Step 2: Rotate \(\hat{j}\)
(-\(\sin \theta\), \(\cos \theta\))
Original \(\hat{j} = (0, 1)\).
The Second Column
\(\begin{bmatrix} -\sin(\theta) \\ \cos(\theta) \end{bmatrix}\)
The Master Formula
R(\(\theta\)) =
[
\(\cos \theta\)
\(-\sin \theta\)
\(\sin \theta\)
\(\cos \theta\)
]
Angle 90°
\(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\)
Angle 180°
\(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
Angle 270°
\(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)
Rotating Shapes Workshop Rotating Real Shapes
WORKSHOP 03: ROTATION MATRICES
Name: __________________________ Date: __________________________
The Rotation Matrix
Rotate any vector by angle \(\theta\) counter-clockwise about the origin.
\( R(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \)
Part 1: The 45° Spin
Let's rotate the vector \(\vec{v} = \begin{bmatrix} 2 \\ 0 \end{bmatrix}\) by 45 degrees. Recall: \(\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707\)
Set up the multiplication:
[
0.707
0.707
-0.707
0.707
] [
2
0
] =
Final Vector Output:
[ \(\dots\) , \(\dots\) ]
Sketch Your Result
Part 2: The "Starship" Maneuver
A simple triangle (starship) has vertices at \(A(0,0)\), \(B(2,0)\), and \(C(1,2)\). We want to rotate the ship 90 degrees CCW about the origin.
Vertex Original Position \(\vec{v}\) Rotation Matrix \(R(90^\circ)\) New Position \(\vec{v}'\) A (0, 0) \(\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) ( \(\dots\) , \(\dots\) ) B (2, 0) ( \(\dots\) , \(\dots\) ) C (1, 2) ( \(\dots\) , \(\dots\) )
Visualization: Plotting the Starship
Before Rotation
Grid units: 1
After 90° CCW Rotation
Plot your calculated points A', B', and C' above.
THE INVERSE CHALLENGE
A video game character is currently at \((5, 5)\). The computer needs to rotate the character around the origin so it lands at \((-5, 5)\).
1. Identify the angle:
What angle is needed to move from quadrant I to quadrant II?
2. Construct the Rotation Matrix:
[
?
?
?
?
]
3. Thinking deeper:
Will a rotation matrix ever change the distance of the character from the origin? Why or why not?
Order Matters Slides Lesson 4
Order Matters
Why Matrix Multiplication is actually a "chain of events" in geometric space.
01
Reflect then Shear
02
Shear then Reflect
Do we land in the same spot? Let's find out.
Composition of Functions
THE "CHAIN"
If we apply Transformation A then Transformation B ...
\(\vec{v}\)
Input
A
Step 1
B
Step 2
B(A(\(\vec{v}\)))
Final Result
\( \vec{v}_{final} = (B \cdot A) \vec{v} \)
Read from right to left! The first transformation is closest to the vector.
Visual Proof: \(A \cdot B \neq B \cdot A\)
A: Rotation \(\rightarrow\) B: Translation
"Rotate about origin, then shift right."
B: Translation \(\rightarrow\) A: Rotation
"Shift right, then rotate about origin."
The order of events changes the destination!
Graphics Pipelines
In games like Minecraft or Fortnite, your character is a collection of 3D vertices.
The computer doesn't transform them one by one. It creates one "Mega Matrix" (Composition) that does everything at once.
M
=
P · V · T · R · S
S
Scale
R
Rotate
T
Translate
P
Project
Sequence Scandal Activity The Sequence Scandal
ACTIVITY 04: COMPOSITION
Name: __________________________ Date: __________________________
Case Study
"I performed a horizontal shear (k=1) and then a reflection across the x-axis," said Scientist A.Goal: Determine if they both ended up with the same final matrix.
Matrix S (Shear)
\(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)
Matrix R (Reflect)
\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
Scientist A: Reflect (R) THEN Shear (S)
Formula: \(M_A = S \cdot R\)
[
] \(\cdot\) [
] = [
]
Scientist B: Shear (S) THEN Reflect (R)
Formula: \(M_B = R \cdot S\)
[
] \(\cdot\) [
] = [
]
Conclusion
Are the final matrices identical? YES / NO
Explain geometrically why this result makes sense. Think about where \(\hat{j}\) lands in each scenario.
Vector Velocity Project Guide Vector Velocity
The Graphics Animation Project
Status
ENCODING...
THE CHALLENGE
In modern computer graphics, every movement of a character is calculated frame-by-frame using matrices. Your mission is to animate a polygon "Logo" across a coordinate plane. You must design a series of three distinct animation frames , calculating the exact matrices required to achieve the movement.
Requirements
Frame 1: Rotation around the origin (min 30°)
Frame 2: Non-uniform Dilation (Stretch/Squish)
Frame 3: A Shear or a Composite Transformation
INPUT DATA
Starting Logo Vertices:
A (1, 1)
B (3, 1)
C (2, 3)
Animation Planning Log
Frame 01: The Spin
Action: Rotate logo by __________ degrees.
[Matrix \(R_1\)]
[ ___ , ___ ]
New Vertices: A' B' C'
Frame 02: The Morph
Action: Stretch horizontally by __________.
[Matrix \(M_2\)]
[ ___ , ___ ]
New Vertices: A'' B'' C''
Frame 03: The Finish
Action: Final Transformation.
[Matrix \(M_3\)]
[ ___ , ___ ]
Final Vertices
Visual Sequence Export
Plot each frame of your animation on the grids below. Use arrows to show the direction of movement between frames.
Initial State (Frame 0)
After Frame 1
After Frame 2
Final Destination (Frame 3)
The "Mega Matrix" (Composition)
Instead of performing 3 multiplications, a computer would multiply all your matrices together first. Write out the final product \(M_{total} = M_3 \cdot M_2 \cdot R_1\) and explain how this single matrix represents the entire journey.
[
]
Vector Velocity Rubric Evaluation Rubric
Vector Velocity Animation Project
Criteria Exceptional (4) Proficient (3) Developing (2-1) Mathematical Accuracy All matrix-vector multiplications and composite products are calculated with 100% accuracy. Most calculations are correct; minor arithmetic errors do not impact the overall animation logic. Significant errors in matrix multiplication or vertex coordinate calculations. Transformation Application Successfully applies 3+ distinct transformations, including correct use of rotation and non-uniform scaling. Applies 3 transformations; may have slight conceptual confusion with one type (e.g., shear). Missing required transformation types or misidentifies transformation effects. Visual Alignment Plots perfectly match calculated coordinates. Use of arrows and labels is professional and clear. Plots generally match calculations; some minor graphing inaccuracies. Plots do not correspond to calculated values or are illegible. Conceptual Depth Deep understanding of composition (order of operations) and geometric effects of matrix entries. Shows understanding of how matrices move shapes, with some minor gaps in the "why." Unable to explain the relationship between matrix values and geometric movement.
Strengths
Areas for Growth
Final Score
___ / 16
