Linear Code Slides
Algebra 1 Foundation
8th Grade Math
Unlock The Secret Pattern
Cracking the Code of
y = mx + b
Every straight line follows one simple two-step rule: Begin at your start, then Move with rhythm!
b
Where We BEGIN
The starting point on the vertical y-axis
m
How We MOVE
Rise over Run to the next dot
Real-World Intuition
The Savings Jar Story
Real Life → Math
You start with $5 in your jar.
Each week, you add +$2 to the jar.
Total Money Rule:
Total Money = $2 × (Weeks) + $5
Spot The 2 Key Numbers
$5
The Starting Amount
Where we begin before week 1
+$2
The Constant Change
How much we add each week
“Math gives these two special numbers famous names: b and m!”
In algebra, we write this exact same pattern as: y = 2x + 5
Spot The Pattern
The Savings Table: Week vs. Total
Week → Total Saved
| Week (x) |
Total Saved (y) |
| Week 0 (Start) |
$5 b = 5 |
| Week 1 |
$7 +$2 |
| Week 2 |
$9 +$2 |
| Week 3 |
$11 +$2 |
| Week 4 |
$13 +$2 |
b
Look at Week 0 → Starting Amount ($5)
Before any weeks pass, you already have $5.
In algebra, this is b = 5 (Where we BEGIN)!
m
Look at the Jumps → Adding +$2 Each Week
Every step down the table increases by +$2.
In algebra, this is m = 2 (How we MOVE)!
Put them together into the slope-intercept equation: Total Money (y) = 2x + 5
The Formula
Anatomy of Slope-Intercept Form
Formula Breakdown
y
Output
=
m
SLOPE
x
Input
b
Y-INTERCEPT
b
b = “BEGIN”
The y-intercept. Plot your start on the vertical y-axis at (0, b).
m
m = “MOVE”
The slope. Move to your next point using Rise ÷ Run (Up/Down then Right).
Golden Rule: First we BEGIN on the y-axis, then we MOVE with slope!
Step 1A Deep Dive • Above Zero
Positive Intercept: +b
Start ABOVE (0,0)
Model Equation
y = 2x + 3
The plus sign tells you to climb UP above the origin (0,0).
↑
Start Point Coordinates
(0, +3)
Notice the faint line passing through (0, 3)—that is the path slope creates!
Climbing UP to (0, 3)
x y-axis +3 UP b = +3 (0, 3) (0,0)
Plus sign → Count UP! (Faint dashed line shows the slope path)
Quick Check: In y = 5x + 6, where do you begin? At (0, 6) • UP 6
Step 1B Deep Dive • Below Zero
Negative Intercept: -b
Start BELOW (0,0)
Model Equation
y = 2x - 4
The minus sign tells you to drop DOWN below the origin (0,0).
↓
Start Point Coordinates
(0, -4)
Notice the faint line passing through (0, -4)—that is the path slope creates!
Dropping DOWN to (0, -4)
x y-axis (0,0) -4 DOWN b = -4 (0, -4)
Minus sign → Count DOWN! (Faint dashed line shows the slope path)
Quick Check: In y = ½x - 5, where do you begin? At (0, -5) • DOWN 5
Step 2 Deep Dive
Meet “m” : The Slope (Rise / Run)
m = MOVE Rule
How Slope Moves
Slope is a fraction that tells you how to travel to the next dot:
m = RISE (Up / Down)
RUN (Always Right →)
Pro Tip: If m = 3, write it as 3 ÷ 1!
Positive Slope (+)
Uphill from Left to Right
m = +2 = UP 2
3 = RIGHT 3
↗
Think: Puffing UP a hill as you go right!
Negative Slope (-)
Downhill from Left to Right
m = -1 = DOWN 1
4 = RIGHT 4
↘
Think: Skiing DOWN a hill as you go right!
Always RUN to the RIGHT (like reading a book)! Numerator = Up/Down • Denominator = Right
Practice & Model
Worked Examples: Identify “m” and “b”
Skill Builder
1
Standard Form
y = 3x + 2
Slope (m)
3 (Up 3, Right 1)
y-Intercept (b)
(0, 2)
2
Fraction Slope & Negative Start
y = ½x - 4
Slope (m)
½ (Up 1, Right 2)
y-Intercept (b)
(0, -4)
!
Watch Out: The Invisible “1”!
y = -x + 6
Slope (m)
-1 (Down 1, Right 1)
y-Intercept (b)
(0, 6)
Remember: If there is no number right in front of x, the slope is 1 (or -1 if negative)!
The 3-Step Method
Graphing: y = ⅔x - 1
Step-by-Step Model
1
BEGIN: Plot “b” on the y-axis
b = -1. Put your first dot on the y-axis at (0, -1).
2
MOVE: Use slope “m” (Rise / Run)
m = ⅔. Count UP 2 and RIGHT 3. Plot dot #2 at (3, 1)!
3
CONNECT: Draw the Line with Arrows
Line up your ruler across both dots, draw a clean straight line, and put arrows on each end!
x y +2 Up +3 Right (0,-1) (3,1)
• Dot 1: Begin (0, -1) • Dot 2: Move to (3, 1)
Only two dots are needed to lock down any straight line!
Your Turn • Partner Showdown
Can You Crack This Line?
Turn & Talk
Target Equation
y = -2x + 4
Q1
Where do you BEGIN?
What are the coordinates of your start point on the y-axis?
Answer: (0, 4)
Q2
How do you MOVE?
Write -2 as Rise/Run. Where do you travel from dot #1?
Answer: Down 2, Right 1
Q3
Uphill or Downhill?
Looking left to right, does this line rise up or fall down?
Answer: Downhill (↘)
Linear Code Mastered:
• b = Begin on y-axis • m = Move with Rise/Run • Connect the Dots!