HavenHub Math • Edition 10 • Student Workbook

Unit 2: The Mountain

"Mastering the Rate of Change"

The Climber's Meditation: Counting the Cost

When we look at a mountain, we see its steepness. In mathematics, we call this Slope. It is the measure of effort. It tells us how much we must lift our feet (Rise) for every step forward (Run).

Jesus told His disciples to "count the cost" of following Him (Luke 14:28). Learning slope is learning to count the cost of the path. Whether the path is steep or gentle, uphill or downhill, we can measure it with truth. We do not fear the mountain; we calculate it.

Lesson 2.1 & 2.2: The Ascent (Rise and Run)

The Law of the Slope: Slope ($m$) = Rise / Run

The Rise (Vertical Change) sits on the throne. The Run (Horizontal Change) supports it. Never flip the fraction!
1. The Ratio of Steepness: Write the fraction.
A. Rise 10, Run 2

$m = \frac{10}{2} = $
B. Rise 4, Run 4

$m = \frac{4}{4} = $
C. Rise 1, Run 3

$m = $
D. Rise 0, Run 5

$m = $
2. Grid Count: Find the Slope from the Picture.

Count the boxes UP and OVER between the two dots.

Rise: Run: Slope:

Lesson 2.3: Direction of the Path (Positive vs. Negative)

The Rule of Reading: Always read the line from LEFT to RIGHT.
Going Up? Positive (+).
Going Down? Negative (-).
1. Climb or Slide? Label the slope type.

Type:

Type:

2. Discernment: Understanding Negative Numbers.

A slope of $-\frac{1}{2}$ means we go DOWN 1 and RIGHT 2.

Draw a quick sketch of a negative slope below.

Lesson 2.4: The Floor and the Wall (Zero and Undefined)

The Extremes:
Zero Slope (0): A flat floor. You can walk on it forever without effort.
Undefined Slope: A vertical wall. You cannot walk on it. It is impossible (Division by Zero).
1. Identify the Special Case.

Slope:

Slope:

The Scribe's Precision Grid: 100 Slope Challenges

Calculate the slope ($m$) or identify the direction. Assume Rise/Run format.

1. Rise 2, Run 1 $\rightarrow m=$
2. Rise 4, Run 2 $\rightarrow m=$
3. Rise 10, Run 5 $\rightarrow m=$
4. Rise 3, Run 3 $\rightarrow m=$
5. Rise 0, Run 5 $\rightarrow m=$
6. Rise 5, Run 0 $\rightarrow m=$
7. Rise -2, Run 1 $\rightarrow m=$
8. Rise -10, Run 2 $\rightarrow m=$
9. Rise 1, Run 2 $\rightarrow m=$
10. Rise 1, Run 3 $\rightarrow m=$
11. $10/2$ simplified is...
12. $100/10$ simplified is...
13. $5/5$ simplified is...
14. $-5/5$ simplified is...
15. $0/10$ simplified is...
16. $10/0$ is...
17. Uphill is (Pos/Neg)?
18. Downhill is (Pos/Neg)?
19. Flat is (Zero/Undef)?
20. Vertical is (Zero/Undef)?
21. Rise 6, Run 2 $\rightarrow 3$
22. Rise 8, Run 4 $\rightarrow 2$
23. Rise 12, Run 3 $\rightarrow 4$
24. Rise 20, Run 2 $\rightarrow 10$
25. Rise 100, Run 1 $\rightarrow 100$
26. Rise 1, Run 100 $\rightarrow 0.01$
27. Rise -4, Run 2 $\rightarrow -2$
28. Rise -6, Run 2 $\rightarrow -3$
29. Rise -8, Run 2 $\rightarrow -4$
30. Rise -10, Run 5 $\rightarrow -2$
31. Grade 100/100 $\rightarrow 1$
32. Grade 50/100 $\rightarrow 0.5$
33. Pitch 4/12 $\rightarrow 1/3$
34. Pitch 6/12 $\rightarrow 1/2$
35. Pitch 12/12 $\rightarrow 1$
36. Steep: 10 or 2? (10)
37. Flat: 10 or 0.1? (0.1)
38. Steep: -10 or -2? (-10)
39. Flat: -10 or -0.1? (-0.1)
40. $y=2x \rightarrow m=2$
41. $y=5x \rightarrow m=5$
42. $y=-3x \rightarrow m=-3$
43. $y=x \rightarrow m=1$
44. $y=-x \rightarrow m=-1$
45. $y=0.5x \rightarrow m=0.5$
46. $(1,1)$ to $(2,2) \rightarrow 1$
47. $(1,1)$ to $(3,3) \rightarrow 1$
48. $(0,0)$ to $(2,4) \rightarrow 2$
49. $(0,0)$ to $(3,6) \rightarrow 2$
50. $(0,0)$ to $(5,10) \rightarrow 2$
51. $(0,0)$ to $(2,-4) \rightarrow -2$
52. $(0,0)$ to $(3,-6) \rightarrow -2$
53. $(1,5)$ to $(2,5) \rightarrow 0$
54. $(5,1)$ to $(5,2) \rightarrow Undef$
55. Rate: $10/hr. m=10$.
56. Rate: $20/hr. m=20$.
57. Rate: $5/min. m=5$.
58. Slope is m.
59. m is for Mountain.
60. Rise/Run.
61. Rise/Run.
62. Rise/Run.
63. Don't flip it.
64. Keep it upright.
65. Y on top.
66. X on bottom.
67. Ladder on floor.
68. Sky on ground.
69. Top is $\Delta y$.
70. Bottom is $\Delta x$.
71. $1/2$ is gentle.
72. $2/1$ is steep.
73. $10/1$ is hard.
74. $100/1$ is wall-like.
75. $-1/2$ is easy slide.
76. $-2/1$ is fast slide.
77. $-10/1$ is dangerous.
78. $0/1$ is nap time.
79. $1/0$ is broken.
80. 1.00 Shalom.
81. Is slope a fraction? (Yes)
82. Can slope be a decimal? (Yes)
83. Can slope be zero? (Yes)
84. Can slope be infinite? (Yes/Undef)
85. Is slope constant on a line? (Yes)
86. Does slope change on a curve? (Yes)
87. We study lines. (Yes)
88. We are climbers. (Yes)
89. We calculate. (Yes)
90. We master. (Yes)
91. $(0,0)$ start.
92. Up 1, Over 1.
93. Up 2, Over 2.
94. Up 3, Over 3.
95. Down 1, Over 1.
96. Down 2, Over 2.
97. Left to Right.
98. Always Forward.
99. Never Back.
100. 1.00 Shalom.

The Prophet's Vision: Real-World Rates

1. The Carpenter's Roof: A roof rises 6 feet for every 12 feet of run. What is the pitch (slope) of the roof? Simplify your answer.
$m = $
2. The Driver's Speed: A car travels 120 miles in 2 hours. What is the rate of speed (slope)? Rise is miles, Run is hours.
$m = $ mph
3. The Sinking Ship: A ship is taking on water. The water level rises 4 inches every 10 minutes. What is the rate of flooding?
$m = $ inches/min

The Artist's Studio: Slope Art

The Challenge: Draw lines with specific slopes to create a mountain scene. Start each line from the previous endpoint.
1. The Twin Peaks:
Start at (-5, 0)
$\rightarrow$ Draw a line with Slope $m=2$ for 2 units right. (End at -3, 4)
$\rightarrow$ Draw a line with Slope $m=-2$ for 2 units right. (End at -1, 0)
$\rightarrow$ Draw a line with Slope $m=3$ for 1 unit right. (End at 0, 3)
$\rightarrow$ Draw a line with Slope $m=-3$ for 1 unit right. (End at 1, 0)
$\rightarrow$ Draw a line with Slope $m=0$ connecting (-5, 0) to (1, 0).

Unit 1 Review: The Map

Recall the Grid: Slope happens ON the Coordinate Plane. Don't forget where you are!
1. Plot the Points: $(2, 3)$ and $(-1, -4)$.
2. Identify the Quadrant: Where is $(-5, 2)$?

Answer: Quadrant

Echad Extension: The Zero and the Infinite

In math, Zero and Undefined are opposites, but they are related. A slope of zero is perfect rest (The Sabbath Floor). A slope of undefined is perfect power (The Divine Wall).

Reflection: Why can we walk on zero but not on undefined? How does this relate to the difference between "Human Effort" (which has limits) and "God's Power" (which has no limit)?

The Scribe's Journal: Reflections on Unit 2

Meditation: The Struggle Upward

You have learned that slope is a measure of steepness. In life, some days are steeper than others. But a steep slope means you are gaining height quickly.

Reflect: Think of a "mountain" you are facing right now (a hard task, a difficult skill). What is the "Rise" (the effort) and what is the "Run" (the time)? Write about your climb.

The Scribe's Dictionary: Edition 10 Vocabulary

Slope ($m$):
The measure of the steepness and direction of a line.
Rise:
The vertical change (Change in Y). The numerator of the slope fraction.
Run:
The horizontal change (Change in X). The denominator of the slope fraction.
Positive Slope:
A line that moves UP from left to right.
Negative Slope:
A line that moves DOWN from left to right.
Zero Slope:
A horizontal line. $m = 0$.
Undefined Slope:
A vertical line. The slope cannot be calculated because you cannot divide by zero.

The Scribe's Dictionary: Unit 2 Vocabulary

Slope ($m$):
The measure of the steepness and direction of a line. Calculated as Rise divided by Run.
Rise:
The vertical change between two points (Change in Y). Positive is Up, Negative is Down.
Run:
The horizontal change between two points (Change in X). Always read left-to-right.
Rate of Change:
How fast one quantity changes in relation to another (e.g., Miles per Hour).
Linear Equation:
An equation that makes a straight line when graphed.
Positive Slope:
A line that goes UPHILL from left to right.
Negative Slope:
A line that goes DOWNHILL from left to right.
Zero Slope:
A horizontal line ($y = 5$). It has no steepness.
Undefined Slope:
A vertical line ($x = 5$). It is impossible to walk on.

The Scribe's Precision Grid: 60 Slope Drills

Calculate the slope ($m$) or identify the type.

1. Rise 4, Run 2 $\rightarrow m=$
2. Rise 6, Run 3 $\rightarrow m=$
3. Rise 10, Run 2 $\rightarrow m=$
4. Rise 5, Run 5 $\rightarrow m=$
5. Rise 1, Run 2 $\rightarrow m=$
6. Rise 1, Run 3 $\rightarrow m=$
7. Rise 0, Run 4 $\rightarrow m=$
8. Rise 4, Run 0 $\rightarrow m=$
9. Rise -2, Run 1 $\rightarrow m=$
10. Rise -4, Run 2 $\rightarrow m=$
11. Rise -10, Run 5 $\rightarrow m=$
12. Rise -3, Run 3 $\rightarrow m=$
13. Rise 100, Run 100 $\rightarrow m=$
14. Rise 50, Run 1 $\rightarrow m=$
15. Rise 1, Run 50 $\rightarrow m=$
16. $(1,1)$ to $(2,3) \rightarrow m=$
17. $(0,0)$ to $(4,4) \rightarrow m=$
18. $(0,0)$ to $(2,1) \rightarrow m=$
19. $(1,5)$ to $(1,10) \rightarrow m=$
20. $(5,1)$ to $(10,1) \rightarrow m=$
21. Uphill $\rightarrow$ Type:
22. Downhill $\rightarrow$ Type:
23. Flat $\rightarrow$ Type:
24. Vertical $\rightarrow$ Type:
25. $y=2x \rightarrow m=$
26. $y=5x \rightarrow m=$
27. $y=-x \rightarrow m=$
28. $y=x \rightarrow m=$
29. $y=1/2x \rightarrow m=$
30. $y=-2x \rightarrow m=$
31. Rise 3, Run 1 $\rightarrow$
32. Rise 6, Run 2 $\rightarrow$
33. Rise 9, Run 3 $\rightarrow$
34. Rise 12, Run 4 $\rightarrow$
35. Rise 15, Run 5 $\rightarrow$
36. Steep: 5 or 1?
37. Steep: 1/2 or 1/4?
38. Flat: 0 or 10?
39. Steep: -10 or -1?
40. Flat: -0.1 or -5?
41. Grade 100% $\rightarrow m=$
42. Grade 50% $\rightarrow m=$
43. Grade 0% $\rightarrow m=$
44. Speed 60mph $\rightarrow m=$
45. Speed 30mph $\rightarrow m=$
46. Rise 2, Run 2 $\rightarrow$
47. Rise 3, Run 3 $\rightarrow$
48. Rise 4, Run 4 $\rightarrow$
49. Rise 5, Run 5 $\rightarrow$
50. Rise 100, Run 100 $\rightarrow$
51. Rise -2, Run 2 $\rightarrow$
52. Rise -3, Run 3 $\rightarrow$
53. Rise -4, Run 4 $\rightarrow$
54. Rise -5, Run 5 $\rightarrow$
55. Rise -10, Run 10 $\rightarrow$
56. $10/2$ $\rightarrow$
57. $2/10$ $\rightarrow$
58. $5/1$ $\rightarrow$
59. $1/5$ $\rightarrow$
60. $0/5$ $\rightarrow$

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

A student says: "The slope is $Run/Rise$ because you walk before you fly."

Is this the Truth? Why is Slope always $\Delta y / \Delta x$? Think about a roof. If the Run was on top, would a flat roof have a big slope or a small slope? Correct the student's formula.

Verdict:

The Covenant

"I, a Climber of the Mountain of Math, promise to calculate my steps with care. I will honor the Rise and the Run. I will not fear the steep path, nor will I be careless on the slide. I know that God is the Lord of the Hills and the Valleys."

Signed:

Date:

UNIT 2 MASTERED!

You have conquered the steepness.

HALLELUYAH!

⛰️