Pre-Algebra bridges arithmetic (working with numbers) and algebra (working with variables and equations). You'll learn to think abstractly about mathematical relationships.
Key Topics Covered:
Integers and the number line
Operations with negative numbers
Variables and expressions
Solving one-step equations
Ratios, proportions, and percentages
The coordinate plane
Introduction to functions
"But let all things be done decently and in order." - 1 Corinthians 14:40
Mathematics reflects Yahuah's orderly creation!
1Integers and the Number Line
What Are Integers?
Integers are whole numbers that can be positive, negative, or zero.
Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...
NOT integers: Fractions (1/2) or decimals (3.5)
The Number Line
←─────────────────────────────────────────→
-5 -4 -3 -2 -1 0 1 2 3 4 5
← Negative | Zero | Positive →
Key Concepts
Positive numbers: Greater than zero (to the right)
Negative numbers: Less than zero (to the left)
Opposites: Same distance from zero, different sides (3 and -3)
Absolute value: Distance from zero, always positive. |−5| = 5
Real-World Examples
Temperature: 10°F above zero = +10; 10°F below zero = -10
Money: $50 in savings = +50; $50 debt = -50
Elevation: 100 ft above sea level = +100; 100 ft below = -100
Time: 3 years before Messiah's birth = -3 (BC); 3 years after = +3 (AD)
"And Elohim called the light Day, and the darkness he called Night." - Genesis 1:5
Just as Yahuah made distinctions (light/dark), math distinguishes positive from negative!
Practice Problems
1
Plot these integers on a number line: -3, 5, -1, 0, 2
2
Find the absolute value:
a) |−7| =
b) |12| =
c) |−15| =
d) |0| =
3
Write the opposite of each integer:
a) 8 →
b) -4 →
c) -12 →
d) 0 →
4
Compare using < or >:
a) -5 3
b) -2 -7
c) 0 -4
d) -10 -3
Parent Check
2Adding and Subtracting Integers
Rules for Adding Integers
Same Signs: Add the absolute values, keep the sign
5 + 3 = 8 | (-5) + (-3) = -8
Different Signs: Subtract absolute values, take sign of larger
5 + (-3) = 2 | (-5) + 3 = -2
Example: Adding Integers
Solve: (-7) + (-4)
Step 1: Both negative, so add absolute values: 7 + 4 = 11
Step 2: Keep the negative sign: -11
Answer: (-7) + (-4) = -11
Rules for Subtracting Integers
Key Rule: Subtracting is the same as adding the opposite!
a - b = a + (-b)
5 - 8 = 5 + (-8) = -3
(-3) - (-7) = (-3) + 7 = 4
Example: Subtracting Integers
Solve: (-6) - (-9)
Step 1: Change to addition of opposite: (-6) + 9
Step 2: Different signs, subtract: 9 - 6 = 3
Step 3: Take sign of larger absolute value: positive
Answer: (-6) - (-9) = 3
"For where your treasure is, there will your heart be also." - Matthew 6:21
If you have $10 (+10) and spend $15, you're in debt: 10 + (-15) = -5
Practice Problems
1
Add:
a) 8 + (-3) =
b) (-5) + (-7) =
c) (-9) + 15 =
d) 6 + (-6) =
2
Subtract:
a) 7 - 12 =
b) (-8) - 5 =
c) (-4) - (-10) =
d) 3 - (-7) =
3Word Problem: The temperature was -5°F in the morning. By afternoon, it rose 12 degrees. What was the afternoon temperature?
4Word Problem: A submarine at -200 feet dives another 75 feet. What is its new depth?
Parent Check
3Multiplying and Dividing Integers
Sign Rules for Multiplication and Division
Signs
Result
Example
(+) × (+)
Positive
3 × 4 = 12
(−) × (−)
Positive
(-3) × (-4) = 12
(+) × (−)
Negative
3 × (-4) = -12
(−) × (+)
Negative
(-3) × 4 = -12
Same signs = Positive | Different signs = Negative
(Same rules apply to division!)
Example: Multiplying Integers
Solve: (-8) × (-5)
Step 1: Multiply the absolute values: 8 × 5 = 40
Step 2: Determine the sign: negative × negative = positive
Answer: (-8) × (-5) = 40
Example: Dividing Integers
Solve: (-36) ÷ 6
Step 1: Divide the absolute values: 36 ÷ 6 = 6
Step 2: Determine the sign: negative ÷ positive = negative
Answer: (-36) ÷ 6 = -6
"Be fruitful, and multiply." - Genesis 1:28
Multiplication increases - or decreases, with negatives!
Practice Problems
1
Multiply:
a) 7 × (-6) =
b) (-9) × (-4) =
c) (-5) × 8 =
d) (-3) × (-3) × (-3) =
2
Divide:
a) (-48) ÷ 8 =
b) 56 ÷ (-7) =
c) (-45) ÷ (-9) =
d) (-100) ÷ 25 =
3
Mixed operations:
a) (-4) × 5 + (-10) =
b) 24 ÷ (-6) - 3 =
c) (-2) × (-3) × (-4) =
Parent Check
4Variables and Expressions
What is a Variable?
A variable is a letter that represents an unknown number.
Common variables: xyna
Algebraic Expressions
An expression combines numbers, variables, and operations.
3x means "3 times x" (3 × x)
x + 5 means "x plus 5"
2x - 7 means "2 times x, minus 7"
x/4 means "x divided by 4"
Evaluating Expressions
To evaluate an expression, substitute a value for the variable.
If x = 4, then 3x + 2 = 3(4) + 2 = 12 + 2 = 14
Example: Evaluating Expressions
Evaluate 5x - 3 when x = 7
Step 1: Substitute: 5(7) - 3
Step 2: Multiply: 35 - 3
Step 3: Subtract: 32
Answer: 5x - 3 = 32 when x = 7
"For You have created all things, and for Your pleasure they are and were created." - Revelation 4:11
Variables let us describe patterns in Yahuah's orderly creation!
Practice Problems
1
Evaluate when x = 5:
a) 4x =
b) x + 9 =
c) 2x - 3 =
d) x² =
2
Evaluate when a = -3:
a) 6a =
b) a + 10 =
c) 2a + 5 =
d) a² =
3
Write as an algebraic expression:
a) Five more than a number:
b) Twice a number:
c) A number decreased by 7:
d) The quotient of a number and 3:
Parent Check
5Solving One-Step Equations
What is an Equation?
An equation is a statement that two expressions are equal.
Example: x + 5 = 12
Goal: Find the value of the variable that makes the equation true.
The Golden Rule of Equations
Whatever you do to one side, you must do to the other!
This keeps the equation balanced, like a scale.
Example 1: Addition Equation
Solve: x + 7 = 15
Step 1: Subtract 7 from both sides: x + 7 - 7 = 15 - 7
Step 2: Simplify: x = 8
Check: 8 + 7 = 15 ✓
Example 2: Multiplication Equation
Solve: 4x = 28
Step 1: Divide both sides by 4: 4x ÷ 4 = 28 ÷ 4
Step 2: Simplify: x = 7
Check: 4(7) = 28 ✓
Inverse Operations
Operation
Inverse
Addition (+)
Subtraction (−)
Subtraction (−)
Addition (+)
Multiplication (×)
Division (÷)
Division (÷)
Multiplication (×)
"A false balance is abomination to Yahuah: but a just weight is His delight." - Proverbs 11:1
Equations must stay balanced - just like Yahuah's justice!
Practice Problems
1
Solve (addition/subtraction):
a) x + 9 = 14 → x =
b) y - 5 = 12 → y =
c) n + 8 = 3 → n =
d) a - 7 = -2 → a =
2
Solve (multiplication/division):
a) 5x = 35 → x =
b) y/4 = 6 → y =
c) -3n = 21 → n =
d) a/(-2) = 8 → a =
3Word Problem: A number plus 12 equals 45. What is the number?
Equation:
Solution:
Parent Check
6Ratios and Proportions
What is a Ratio?
A ratio compares two quantities. It can be written as:
a to b
a : b
a/b (as a fraction)
Example: If there are 3 boys and 5 girls, the ratio of boys to girls is 3:5
What is a Proportion?
A proportion states that two ratios are equal.
Example: 2/3 = 8/12 (both simplify to the same ratio)
Cross Multiplication
To solve a proportion, cross multiply:
If a/b = c/d, then a × d = b × c
Example: Solving a Proportion
Solve: 3/5 = x/20
Step 1: Cross multiply: 3 × 20 = 5 × x
Step 2: Simplify: 60 = 5x
Step 3: Divide by 5: x = 12
Check: 3/5 = 12/20 = 0.6 ✓
"A tithe of all" - Leviticus 27:30
A tithe is a ratio: 1/10 or 1:10 - one part out of every ten belongs to Yahuah!
Practice Problems
1
Write each ratio in simplest form:
a) 12:16 =
b) 15/25 =
c) 8 to 20 =
2
Solve each proportion:
a) 2/5 = x/15 → x =
b) 4/7 = 12/n → n =
c) x/8 = 9/12 → x =
3Word Problem: If 3 apples cost $2, how much do 12 apples cost?
Proportion:
Answer:
4Tithe Problem: If your income is $150, what is the tithe (1/10)?
Parent Check
7Percentages
What is a Percentage?
A percent means "per hundred" or "out of 100."
25% means 25 out of 100, or 25/100 = 0.25
Converting Between Forms
Percent
Decimal
Fraction
50%
0.50
1/2
25%
0.25
1/4
10%
0.10
1/10
75%
0.75
3/4
100%
1.00
1
Finding a Percent of a Number
Part = Percent × Whole
Convert percent to decimal first!
Example: 20% of 80 = 0.20 × 80 = 16
Example: Percent Problems
Problem: What is 15% of 60?
Step 1: Convert to decimal: 15% = 0.15
Step 2: Multiply: 0.15 × 60 = 9
Answer: 15% of 60 = 9
"And all the tithe of the land... is Yahuah's: it is holy unto Yahuah." - Leviticus 27:30
A tithe is 10% - giving back to Yahuah from what He has blessed us with!
Practice Problems
1
Convert to decimal:
a) 35% =
b) 8% =
c) 125% =
2
Find the percent:
a) 50% of 84 =
b) 20% of 45 =
c) 10% of 230 =
d) 75% of 120 =
3Tithe Calculation: Calculate the tithe (10%) for each amount:
a) $85 tithe =
b) $120 tithe =
c) $267 tithe =
4Word Problem: A shirt is on sale for 25% off. If the original price is $40, what is the discount? What is the sale price?
Discount:
Sale price:
Parent Check
8The Coordinate Plane
What is a Coordinate Plane?
A coordinate plane is a grid formed by two perpendicular number lines:
x-axis: horizontal (left-right)
y-axis: vertical (up-down)
Origin: where they cross (0, 0)
Ordered Pairs
Every point is written as (x, y)
x = how far left or right | y = how far up or down
Example: (3, 2) means go right 3, up 2
Example: (-2, 4) means go left 2, up 4
The Four Quadrants
Quadrant
Signs
Example
I (upper right)
(+, +)
(3, 2)
II (upper left)
(−, +)
(-3, 2)
III (lower left)
(−, −)
(-3, -2)
IV (lower right)
(+, −)
(3, -2)
"He has made every thing beautiful in His time." - Ecclesiastes 3:11
The coordinate plane helps us map and appreciate Yahuah's ordered creation!
Practice Problems
1
Identify the quadrant for each point:
a) (5, 3) = Quadrant
b) (-4, 2) = Quadrant
c) (-1, -6) = Quadrant
d) (2, -5) = Quadrant
2
Plot these points on a coordinate plane:
A(2, 4), B(-3, 1), C(-2, -3), D(4, -2), E(0, 3)
3
Give the coordinates of a point in each quadrant:
Quadrant I:
Quadrant II:
Quadrant III:
Quadrant IV: