Blueprint Intro Slides Lesson 1.1: Grid Logic
The Matrix Blueprint
Organizing chaotic data into powerful mathematical structures.
What is a Matrix?
A matrix is a rectangular array of numbers organized into rows and columns.
Plural: Matrices
Used to store and manipulate massive datasets simultaneously.
[
2-5 012
]
The Dimensions (Order)
Rows × Columns
"Down then Across"
[ 4 -1 8 ]
1 × 3 Matrix
1 Row, 3 Columns
20
-95
2 × 2 Matrix
2 Rows, 2 Columns
Locating Elements
Each number in a matrix is an element. We find it using its address:
Element aij
i = Row #
j = Column #
[
104-2 078
]
a11 = 10
From Chaos to Structure
Cafeteria Sales Table
Item Mon Tue Pizza 45 52 Tacos 30 65 Salad 12 15
[
4552 3065 1215
]
3 × 2 Matrix
Data Architect Worksheet Data Architect Worksheet
PROJECT: MATRIX BLUEPRINT // UNIT 1.1
Student:
Date:
Rule 1: Dimensions
"Down then Across"
Rows × Columns
Rule 2: Elements
Address by Subscript
aij (Row i, Col j)
1 Identify Dimensions and Elements
A =
5-210 084
1. Dimensions:
2. a21 =
3. a13 =
B =
12 -1 7
4. Dimensions:
5. b21 =
6. b31 =
2 The Blueprint: Table to Matrix
The following table shows the inventory of three types of sneakers (Air, Zoom, Max) across two different retail locations. Convert this data into a matrix I .
Location Air Zoom Max Downtown 45 30 12 Uptown 22 15 40
I =
7. Order of I :
8. Value of i12:
9. Value of i23:
10. What does i21 represent?
3 Reverse Engineering
Given matrix K = [ 4 8 2 ], create a real-world scenario that this matrix could represent. Define what the rows and columns signify.
Blueprint Answer Key Teacher Guide & Answer Key
UNIT 1.1: THE MATRIX BLUEPRINT
Learning Objectives
Identify matrix dimensions (Order) as Rows × Columns.
Locate specific elements using aij subscript notation.
Translate structured tabular data into matrix form.
Common Misconceptions
Mixing up Rows vs Columns (thinking "across then down").
Treating aij as (x, y) coordinates where x is horizontal.
Worksheet Answer Key
Part 1: Identification
Matrix A (2 × 3)
1. Dimensions: 2 × 3
2. a21: 0
3. a13: 10
Matrix B (3 × 1)
4. Dimensions: 3 × 1
5. b21: -1
6. b31: 7
Part 2: Data Synthesis
Matrix I Solution:
453012 221540
7. Order: 2 × 3
8. i12: 30
9. i23: 40
10. i21 Meaning: The number of "Air" sneakers at the Uptown location.
Part 3: Challenge Sample
"The matrix K could represent the number of points scored in three different quarters of a game by a single player. Row = Player, Columns = Quarters 1, 2, and 3."
Discussion Starters
Q: Why are dimensions critical before doing math?
A: Just like you can't add different shaped LEGO blocks, matrix dimensions must 'fit' certain rules for operations to be possible.
Q: Could we transpose the table (swap rows and columns)?
A: Yes! A 2x3 becomes a 3x2. The data is the same, but the "view" changes.
Scaling Grids Slides Lesson 1.2: Grid Logic
Scaling and Shifting
Combining datasets and manipulating values with scalar power.
−
×
Combining Grids
The Golden Rule:
To add or subtract, matrices MUST have the same dimensions.
We simply add or subtract the corresponding elements (same position).
[ 4 2 ] + [ 1 5 ] = [ 5 7 ]
Cannot add a 2x2 and a 1x2.
The Scalar Power
A scalar is just a real number. When we multiply a matrix by a scalar, we scale every single element by that number.
Use Case:
Applying a 20% discount (multiply by 0.8) to an entire price list at once.
3 ×
2 -5
=
6 -15
Warehouse Logic
Warehouse A
[ 100 50 ]
Combine
Warehouse B
[ 80 120 ]
Total Inventory Matrix
[ 180 170 ]
Inventory Shift Worksheet Inventory Shift Worksheet
PROJECT: OPERATIONS & SCALING // UNIT 1.2
Student:
Date:
1 Dimension Verification
Can these matrices be added or subtracted? Circle YES or NO and state why.
[ 4 2 ] ± [ 1 0 ]
YES
NO
Reason:
5-2
± [ 3 8 ]
YES
NO
Reason:
2 Scalar Scaling & Shifts
A. Scalar Multiplication
4
-25 10
=
B. Matrix Subtraction
104
−
7-2
=
3 The Discount Strategy
A electronics store has its current prices stored in Matrix P . They decide to run a "Storewide 25% Off" sale. Multiply Matrix P by a scalar of 0.75 to find the new sale prices.
Original Prices P
$400 $120 $60
0.75 × P =
Critical Thinking: If the store then added a $5 "Shipping Fee" to every item, what matrix operation would you perform next?
Scaling Grids Answer Key Teacher Guide & Answer Key
UNIT 1.2: SCALING AND SHIFTING GRIDS
Learning Objectives
Add/Subtract matrices by identifying matching dimensions.
Apply scalar multiplication to uniform scaling of data.
Recognize scalar multiplication as a distributive property.
Teaching Tip
Remind students that scalar multiplication is like a volume knob—it changes the "magnitude" of every element equally, while matrix addition is like stacking layers of data.
Worksheet Answer Key
Section 1: Dimension Verification
Problem 1: [4 2] ± [1 0]
YES Reason: Both matrices are 1 × 2.
Problem 2: [5, -2]T ± [3 8]
NO Reason: Different dimensions (2 × 1 vs 1 × 2).
Section 2A: Scalar
-820 40
Section 2B: Subtraction
3 6
Section 3: Scenario
$300 $90 $45
Critical Thinking Answer:
"You would add a 3 × 1 matrix where every element is 5 (Matrix Addition)."
Misconception Alert
Watch out for students who multiply only the top-left element by the scalar, or who try to subtract matrices by subtracting the totals of each matrix. Emphasize that position matters.
Multiplication Flow Slides Lesson 1.3: Grid Logic
The Multiplication Flow
Mastering the Row-by-Column dot product algorithm.
Can we multiply?
Matrix A
m × n
Matrix B
n × p
The "Inside" numbers must match!
The product will have the "outside" dimensions: m × p
The Row-by-Column "Dive"
"The Row dives into the Column."
1 Multiply corresponding entries.
2 Add the results (the sum).
3 The sum becomes one element in the product.
[ a b ] ×
x y
= (a × x) + (b × y)
Visualizing the 2x2
12 34
×
56 78
=
Top-Left Element:
(1 × 5) + (2 × 7)
= 19
Multiplication Flow Worksheet Multiplication Flow
PROJECT: DOT PRODUCT MECHANICS // UNIT 1.3
Student:
Date:
1 Can it Flow?
Determine the dimensions of the product matrix. If the product is undefined, write "UNDEFINED".
2 × 3 AND 3 × 4
1 × 2 AND 1 × 2
3 × 2 AND 2 × 1
2 Step-by-Step Dot Product
Multiply the matrices by following the scaffolded calculations below.
1234
×
5678
=
Write final matrix here
Top-Left Element (r1 × c1):
(1 × 5) + (2 × 7) =
Top-Right Element (r1 × c2):
(1 × 6) + (2 × 8) =
Bottom-Left Element (r2 × c1):
(3 × ) + (4 × ) =
Bottom-Right Element (r2 × c2):
Fill this in entirely...
3 Unassisted Calculation
20-1
×
35
=
Multiplication Flow Answer Key Teacher Guide & Answer Key
UNIT 1.3: THE MULTIPLICATION FLOW
Mechanical Focus
This lesson is about accuracy . Students often struggle with the "rhythm" of multiplication. Encourage them to use their fingers: left hand moves across the row, right hand moves down the column.
Common Slip-ups
Multiplying elements in the same position (like addition).
Forget to sum the products (just listing 5 and 14 instead of 19).
Worksheet Answer Key
Section 1: Dimension Check
2 × 4
Undefined
3 × 1
Section 2: Guided Practice
1922 4350
TL: 1(5) + 2(7) = 5 + 14 = 19
TR: 1(6) + 2(8) = 6 + 16 = 22
BL: 3(5) + 4(7) = 15 + 28 = 43
BR: 3(6) + 4(8) = 18 + 32 = 50
Section 3: Unassisted Calculation
610 00 -3-5
Dimensions check: (3×1) times (1×2) = (3×2). Row 1 [2] × [3, 5] = [6, 10] Row 2 [0] × [3, 5] = [0, 0] Row 3 [-1] × [3, 5] = [-3, -5]
Order and Identity Slides Lesson 1.4: Grid Logic
Order and Identity
Breaking the rules of normal math and finding the "One" matrix.
AB ≠ BA Order Matters
AI = A Identity
The Rule Breaker
In normal math, 2 × 3 = 3 × 2. We call this the Commutative Property.
Warning:
AB ≠ BA
Matrix multiplication is NOT commutative.
A × B results in:
[ 10 -2 ]
B × A results in:
[ 4 12 ]
"Swapping the order changes the 'path' the divers take through the pools."
The Identity (I)
The Identity Matrix acts like the number "1" in normal multiplication.
A × I = A
Must be Square (nxn)
1s on the diagonal, 0s everywhere else.
[
100 010 001
]
Why does order matter?
Case A: [Quantity] × [Price]
Works! This gives you the Total Cost.
Case B: [Price] × [Quantity]
Often undefined or gives nonsensical data based on dimension rules.
"In matrix-land, context defines the sequence."
Identity Investigation Worksheet Identity & Order
PROJECT: PROPERTY INVESTIGATION // UNIT 1.4
Student:
Date:
1 Lab: Does Order Matter?
Given matrices A and B , calculate the products in both directions. Use the grids provided for your scratch work.
A =
2104
B =
-1352
Operation 1: A × B
Operation 2: B × A
Conclusion:
Did you get the same result for both?
Does the commutative property (AB = BA) hold true for matrices?
2 Finding the "Identity"
In normal arithmetic, x × 1 = x . In matrix algebra, we use the Identity Matrix (I) .
Multiply Matrix M by Identity Matrix I:
7-3210
×
1001
=
The Result:
3 Blueprint an Identity
Draw the 3 × 3 Identity Matrix below.
[ ]
Order and Identity Answer Key Teacher Guide & Answer Key
UNIT 1.4: ORDER AND IDENTITY
Conceptual Goal
Moving beyond mechanics to properties . The main takeaway is that matrix algebra is a distinct system with its own rules—specifically the lack of commutativity.
The "Aha!" Moment
Watch for students to realize that the Identity Matrix is like a "filter" that lets the original matrix through unchanged. This is essential for understanding matrix inverses later.
Worksheet Answer Key
Section 1: The Commutative Test
A × B Result:
38208
B × A Result:
-2111013
Conclusion: Results are NOT the same. The commutative property does NOT hold.
Section 2: Finding the Identity
7-3210
Result is identical to original Matrix M.
Section 3: Blueprint 3x3 Identity
100 010 001
Challenge Question
"Is it EVER possible for AB = BA?" A: Yes (e.g., if one is Identity, or if they are inverses), but it is not a general rule for all matrices.
Sneaker Empire Slides Lesson 1.5: Grid Logic
Sneaker Empire
Using matrix multiplication to run a multi-million dollar footwear business.
The CEO's Challenge
"We have 3 locations and 4 types of shoes. I need to know the Total Stock Value for each store right now."
Why Matrix Mult?
Calculating this item-by-item takes hours. A single matrix multiplication takes seconds.
Location Matrix 3 × 4
Price Matrix 4 × 1
Total Value 3 × 1
Modeling the Business
Inventory (I)
10155
12820
72215
1004010
Shoes × Store Locations
Prices (P)
$120$85$200$45
Price per Shoe
Analyzing the Product
The result of our I × P multiplication is a single column.
$22,450 $18,900 $14,200
Store 1 (Downtown)
Store 2 (Uptown)
Store 3 (Suburbs)
Sneaker Empire Case Study Sneaker Empire Analysis
PROJECT: DATA-DRIVEN BUSINESS // UNIT 1.5
Student:
Executive Summary
You are the Chief Data Officer for Sole Purpose Inc. , a regional sneaker chain. You manage two locations: The Hub and The Outpost . Your task is to calculate the total inventory value using matrix operations.
1 Step 1: Organize the Inventory (Matrix I)
Current Stock Levels:
Air Max: 50 at Hub, 20 at OutpostZoom: 30 at Hub, 45 at OutpostLegacy: 15 at Hub, 10 at Outpost
Construct Matrix I (2 × 3):
Rows = Locations, Columns = Shoe Types
2 Step 2: Define Price Matrix (Matrix P)
Retail Prices:
Model Price Air Max $150 Zoom $110 Legacy $210
Construct Matrix P (3 × 1):
3 Step 3: Calculate Total Value (I × P)
Show your dot product calculations for both locations.
The Hub Value:
Total: $
The Outpost Value:
Total: $
4 Strategic Expansion
Sole Purpose Inc. is opening a third store, The Peak . It will have exactly 20% more stock of every item than The Hub . What matrix operation would you perform to find the inventory for the third store?
Sneaker Empire Answer Key Teacher Guide & Answer Key
UNIT 1.5: SNEAKER EMPIRE ANALYSIS
Synthesis & Application
This final lesson requires students to integrate every skill learned: organization (Lesson 1), scaling (Lesson 2), mechanics (Lesson 3), and logic (Lesson 5).
Case Study Solutions
Step 1: Matrix I (Inventory)
503015 204510
Dimensions: 2 × 3
Step 2: Matrix P (Prices)
150 110 210
Dimensions: 3 × 1
Step 3: Calculations (I × P)
The Hub (Row 1 × Col 1):
(50 × 150) + (30 × 110) + (15 × 210) = 7500 + 3300 + 3150
$13,950
The Outpost (Row 2 × Col 1):
(20 × 150) + (45 × 110) + (10 × 210) = 3000 + 4950 + 2100
$10,050
Step 4: Strategic Expansion
Operation:
"Scalar Multiplication. Multiply the row vector representing 'The Hub' inventory by a scalar of 1.2 (or perform scalar multiplication of 1.2 × Row 1)."
End of Sequence: Grid Logic Matrix Operations // sole-purpose-analysis-key
