HavenHub Math • Edition 10 • Student Workbook

Unit 3: The Linear Code

"Translating Reality into $y = mx + b$"

The Architect's Meditation: The Faithful Pattern

When we look at a straight line, we are looking at Faithfulness. A line does not wander; it follows a rule. In math, we call this rule the Code. It tells us where we started and how fast we are changing.

"He is before all things, and by him all things consist" (Colossians 1:17). The universe is held together by a consistent logic. As you learn to write the code for a line, remember that you are learning to describe the consistency of God's world.

Lesson 3.1: Deconstructing the Code ($m$ and $b$)

The Formula of the Path:
$y = mx + b$
1. Identify the DNA: Extract the parts from the code.
A. $y = 4x + 7$

$m = $   $b = $
B. $y = -2x - 3$

$m = $   $b = $
C. $y = \frac{1}{3}x + 10$

$m = $   $b = $
D. $y = x - 5$

$m = $   $b = $
2. The Hidden Zero: Spotting the invisible numbers.
A. $y = 5x$

What is the start height ($b$)?

$b = $
B. $y = 8$

What is the rate of change ($m$)?

$m = $

Lesson 3.2 & 3.3: Setting the Anchor and the March

The Graphing Ritual:
  1. Put your pencil on the Origin (0,0).
  2. Move to the Y-Intercept ($b$) and draw a dot. (The Anchor).
  3. From that dot, use the Slope ($m$) to find the next point. (The March).
  4. Connect the dots with a Ruler of Integrity.
1. Executing the Plan: Graph the equation $y = 2x + 1$.

Anchor ($b$) =

March ($m$) =

2. The Deep Start: Graph $y = \frac{1}{2}x - 4$.

Anchor ($b$) =

March ($m$) =

Lesson 3.4: The Codebreaker (Writing the Formula)

Reverse Engineering: To write the code from a picture:
  1. Find where the line hits the Y-axis. This is your $b$.
  2. Pick another clean point and count the Rise and Run to find your $m$.
  3. Plug them into $y = mx + b$.
1. Decode the Path: Write the equation for each line described.
A. The line hits the Y-axis at 5. It goes Up 2 for every 1 step right.

$y = $
B. The line hits the Y-axis at -2. It goes Down 1 for every 3 steps right.

$y = $
C. The line passes through the Origin. Its slope is -4.

$y = $
D. The line is perfectly flat and stays at a height of 10.

$y = $

Lesson 3.5: The Master Architect's Blueprint

1. Full Synthesis: Model and Graph.

A city worker charges a flat fee of $10 just to show up, plus $5 for every hour he works.

If he works for 4 hours, what is the total cost ($y$)?

The Scribe's Precision Grid: 100 Coding Drills

Identify $m$ and $b$ for each equation. Use "0" if a part is missing.

1. $y=2x+3 \rightarrow$
2. $y=x-5 \rightarrow$
3. $y=-3x+10 \rightarrow$
4. $y=1/2x-1 \rightarrow$
5. $y=5x \rightarrow$
6. $y=7 \rightarrow$
7. $y=-x \rightarrow$
8. $y=4x+4 \rightarrow$
9. $y=-2x-2 \rightarrow$
10. $y=10x+100 \rightarrow$
11. $y=0.1x \rightarrow$
12. $y=-0.5x+2 \rightarrow$
13. $y=x \rightarrow$
14. $y=2x \rightarrow$
15. $y=-2x \rightarrow$
16. $y=3x-3 \rightarrow$
17. $y=1/4x+5 \rightarrow$
18. $y=-1/3x-2 \rightarrow$
19. $y=100x \rightarrow$
20. $y=-5 \rightarrow$
21. Start at 2, m=1.
22. Start at 0, m=5.
23. Start at -3, m=2.
24. Start at 10, m=-1.
25. Start at 5, m=0.
26. $y=mx+b$.
27. $y=mx+b$.
28. $y=mx+b$.
29. $y=mx+b$.
30. $y=mx+b$.
31. $y=2x+1$.
32. $y=2x+2$.
33. $y=2x+3$.
34. $y=2x+4$.
35. $y=2x+5$.
36. $y=-x+1$.
37. $y=-x+2$.
38. $y=-x+3$.
39. $y=-x+4$.
40. $y=-x+5$.
41. $y=3x+0$.
42. $y=3x-1$.
43. $y=3x-2$.
44. $y=3x-3$.
45. $y=3x-4$.
46. $y=1/2x+1$.
47. $y=1/2x+2$.
48. $y=1/2x+3$.
49. $y=1/2x+4$.
50. $y=1/2x+5$.
51. $y=10x$.
52. $y=20x$.
53. $y=30x$.
54. $y=40x$.
55. $y=50x$.
56. $y=x+10$.
57. $y=x+20$.
58. $y=x+30$.
59. $y=x+40$.
60. $y=x+50$.
61. Slope is 5.
62. Slope is -2.
63. Slope is 0.
64. Slope is 1.
65. Slope is -1.
66. Int is 4.
67. Int is -2.
68. Int is 0.
69. Int is 10.
70. Int is -1.
71. $y=5x+4$.
72. $y=-2x-2$.
73. $y=0x+0$.
74. $y=x+10$.
75. $y=-x-1$.
76. $(0,5)$ is b.
77. $(0,0)$ is b.
78. $(0,-2)$ is b.
79. $(0,10)$ is b.
80. $(0,1)$ is b.
81. Code is rule.
82. Rule is path.
83. Path is true.
84. True is life.
85. $y=mx+b$.
86. Rise/Run.
87. Y on X.
88. Start on Y.
89. Move on X.
90. Draw the line.
91. Check point 1.
92. Check point 2.
93. Verify code.
94. Halleluyah.
95. Amen.
96. $y=mx+b$.
97. $y=mx+b$.
98. $y=mx+b$.
99. Linear King.
100. 1.00 Shalom.

The Prophet's Vision: The Logic of Providence

1. The Stewardship of Talents: A servant starts with 5 talents. Every year, he earns 2 more through faithful trade. Write the equation for his total talents over time ($x$).
Equation:
2. The Widow's Oil: A jar has 10 cups of oil. Every day, the widow uses 1/2 cup for bread. Write the equation for the remaining oil.
Equation:
3. The City Wall: A wall is 4 feet high. The builders add 3 feet of height every day. How high will the wall be after 6 days? Write the code and solve.
Code: Height:

The Architect's Grid: Construction Drills

The Task: Graph each equation on its own grid. Label the Y-intercept and show the Slope triangle.
1. $y = 3x - 2$
2. $y = -\frac{2}{3}x + 4$
3. $y = x$
4. $y = -5$

Unit 3 Mastery Review: The Deep Dive

The Synthesis: Show that you have mastered the Code.
1. The Mixed Mystery: A line passes through $(0, 5)$ and has a slope of $-2$. Write its code.
$y = $
2. The Origin Path: A line passes through the Origin and the point $(2, 8)$. Find $m$ and $b$, then write the code.
$m = $ $b = $ Code:
3. The Constant Walk: Solve the story. A candle is 12 inches tall. It burns at a rate of 1.5 inches per hour. Write the equation. After how many hours will the candle be gone ($y=0$)?
Code: Time: hours.

The Pattern Seeker: Tables to Equations

The Revelation: Sometimes the code is hidden in a list of numbers. Find the change in $Y$ divided by the change in $X$ to find $m$. Find the $Y$ value when $x=0$ to find $b$.
1. Table A:
xy
02
15
28
311

Start value ($b$):

Growth rate ($m$):

Code:

2. Table B:
xy
010
28
46
64

Start value ($b$):

Growth rate ($m$):

Code:

The Prophet's Vision: Part 2

4. The Shepherd's Flock: A shepherd starts with 50 sheep. He buys 5 more sheep every month. Write the code for his total sheep ($y$) after $x$ months. How many sheep will he have after one year?
Equation: Total Sheep:
5. The Builder's Pay: A stone-cutter is paid $20 per day plus $2 for every block he cuts. Write the code for his daily pay. If he wants to earn $50 in one day, how many blocks must he cut?
Equation: Blocks:
6. The Falling Rain: A barrel starts with 100 gallons of water. It leaks at a rate of 4 gallons per hour. Write the code for the remaining water. How many hours until the barrel is half empty?
Equation: Hours:

The Scribe's Cumulative Review

The Map and the Mountain: Before we leave the Code, let us remember the Grid and the Slope.
1. Location Check (Unit 1): Plot the point $(-4, 2)$ and name the Quadrant.
Quadrant:
2. Slope Check (Unit 2): A line passes through $(0,0)$ and $(3, 6)$. What is its slope?
$m = $
3. Synthesis: If a line has a slope of $2$ and passes through $(0,0)$, what is its equation?

Equation:

4. The Big Picture: If a line is horizontal and passes through the point $(0, -5)$, what is its equation? (Hint: The slope is 0).

Equation:

📋 The Scribe's Final Audit: Unit 3

Before you receive your Signet, you must pass the auditor's check.

  1. Can you identify $m$ and $b$ in $y = -5x + 2$?
  2. Can you graph $y = \frac{1}{2}x - 3$ without starting at the origin?
  3. Can you write the code for a line that starts at 10 and goes up 1 for every 1 step?
  4. Do you understand that a line is just a picture of an equation?
  5. Are you ready to use the Code to solve mysteries in the next unit?

Auditor's Signature:

The Scribe's Journal: Reflections on Unit 3

Meditation: The Anchor and the March

In this unit, you learned that every path has an Anchor ($b$) and a March ($m$). Without the anchor, you don't know where to start. Without the march, you don't know how to move.

Reflect: In your own life, what is your "Anchor"? What is the fixed truth that you start from every day? How does that anchor help you decide which direction to move?

The Scribe's Dictionary: Unit 3 Vocabulary

Slope-Intercept Form:
The equation $y = mx + b$, the universal code for a linear path.
Y-Intercept ($b$):
The point where the line crosses the center wall (the Y-axis). The starting value.
Coefficient:
The number being multiplied by $x$. In this unit, it is the Slope.
Constant:
A number that doesn't change. In this unit, it is the Y-intercept.
Linear:
Following a straight path. A relationship with a constant rate of change.
Modeling:
The act of turning a real-world story into a mathematical code.

C.A.M.E. Verification Node: The Coder's Audit

A student graphs $y = -3x + 4$. They start at 4 on the Y-axis. Then they move UP 3 and RIGHT 1.

Is this the Truth? Look at the sign of the slope. Should the line be climbing or sliding? Correct the student's "March."

Verdict:

The Covenant

"I, a Coder of the Master Line, solemnly promise to follow the blueprint of the equation. I will set my anchor on the Y-axis and march my slope with faithfulness. I will use the code to reveal the hidden consistency of the world, for I know that all things consist in the Truth."

Signed:

Date:

UNIT 3 MASTERED!

You have cracked the code of the path.

HALLELUYAH!

📜