Essential Math Formulas and Concepts: A Quick Reference

Essential Math Formulas and Concepts

Volume of Shapes

(a) Cube:

• Formula: V = s³ (where s is the length of a side)
• Example: A cube with side length 4 cm

V = 4³ = 64 cm³

(b) Cylinder:

• Formula: V = πr²h (where r is the radius and h is the height)
• Example: A cylinder with radius 2 cm and height 6 cm

V = π(2²)(6) = 24π cm³ (approximate value)

Surface Area of Shapes

(a) Cube:

• Formula: SA = 6s² (where s is the length of a side)
• Example: A cube with side length 3 cm

SA = 6(3²) = 54 cm²

(b) Cone:

• Formula: SA = πr(r + √(r² + h²)) (where r is the radius and h is the height)
• Example: A cone with radius 5 cm and height 8 cm

SA = π(5)(5 + √(5² + 8²)) = 129.74 cm² (approximate value)

Stem and Leaf Plot

(a) Mean:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Mean = (5 + 7 + 8 + 9 + 10 + 12 + 15) / 7 = 9

(b) Median:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Median = 9

(c) Mode:

• Example: Data: 5, 7, 8, 9, 9, 9, 10, 12, 15

Mode = 9

(d) Range:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Range = 15 - 5 = 10

(e) Lower Quartile:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Lower Quartile = 7

Box & Whisker Graphs

(a) Inter-Quartile Range (IQR):

• Example: Class A: Q3 = 60, Q1 = 40

IQR = Q3 - Q1 = 60 - 40 = 20

• Class B: Q3 = 80, Q1 = 30

IQR = Q3 - Q1 = 80 - 30 = 50

(b) Comparison:

• Class average (center): Compare the mean of each class.
• Example: Class A mean = 50, Class B mean = 60

Class B has a higher class average.

• Spread of the middle 50%: Compare the IQR values for each class.
• Example: Class A IQR = 20, Class B IQR = 50

Class B has a larger spread of the middle 50%.

Probability

(a) Tree Diagram:

• Example: Tossing a fair coin twice

(b) Probabilities:

• Example: P(H, H) = ¼ (from the tree diagram)
• Example: P(At least one Tail) = ¾ (from the tree diagram)

Probability and Venn Diagram

(a) P(B):

• Example: P(B) = Number of outcomes in B / Total number of outcomes

(b) P(A ∪ B):

• Example: P(A ∪ B) = Number of outcomes in A or B / Total number of outcomes

(c) P(A | B) (conditional probability):

• Example: P(A | B) = Number of outcomes in both A and B / Number of outcomes in B

(d) Independence of events A and B:

• Example: P(A | B) = P(A) (events A and B are independent) or P(A | B) ≠ P(A) (events A and B are not independent)

Solving Quadratic Equations

(a) Solve the quadratic equation y = x² + 7x + 10

• Example: y = x² + 7x + 10

Set y = 0 and solve for x.

x² + 7x + 10 = 0

(x + 5)(x + 2) = 0

x = -5 or x = -2

(b) Solve the quadratic equation y = x(x - 2) - 2

• Example: y = x(x - 2) - 2

Set y = 0 and solve for x.

x(x - 2) - 2 = 0

x² - 2x - 2 = 0

Use the quadratic formula or factorize.

(x + 1 - √3)(x + 1 + √3) = 0

x = -1 - √3 or x = -1 + √3

(c) Solve the quadratic equation y = x² - 6x + 12

• Example: y = x² - 6x + 12

Set y = 0 and solve for x.

x² - 6x + 12 = 0

Use the quadratic formula or factorize.

(x - 3 + √3i)(x - 3 - √3i) = 0

x = 3 + √3i or x = 3 - √3i