Tricks for Memorizing Tables

Table of Contents
1. Foundational Doubling Strategy (Tables 2, 4, 8, 16)
2. Halving Strategy (Table 5 and 15)
3. Tens-Based Anchoring (Tables 10, 20)
4. Neighbor Subtraction Method (Table 9)
5. Neighbor Addition Method (Table 11)
6. Decomposition Strategy (Table 12)
7. Teen Tables (13, 14, 15, 16, 17, 18, 19)
8. Complementary Subtraction (High-Value Tables)
9. Middle-Range Tables (6, 7, 8)
10. Pattern Recognition Systems
11. Advanced Calculation Techniques
12. Strategic Memorization Framework
13. Common Mistakes and Traps
14. Exam-Day Application Strategies
View more

Mastering multiplication tables from 2 to 20 requires strategic shortcuts rather than rote memorization. This compilation presents high-yield tricks and patterns that reduce cognitive load and enable rapid calculation. Each method leverages mathematical properties like doubling, halving, digit sum patterns, and decomposition strategies essential for competitive exam speed.

1. Foundational Doubling Strategy (Tables 2, 4, 8, 16)

These tables form a power-of-2 cascade where each table is double of the previous one. This interconnected pattern allows vertical calculation.

1.1 Core Doubling Chain

• Table of 2: Simply double any number (2 × 7 = 14)
• Table of 4: Double the result of table 2 (4 × 7 = double of 14 = 28)
• Table of 8: Double the result of table 4 (8 × 7 = double of 28 = 56)
• Table of 16: Double the result of table 8 (16 × 7 = double of 56 = 112)

1.2 Practical Application

Example: To find 16 × 9 without memorization:

1. Start with 2 × 9 = 18
2. Double for 4 × 9 = 36
3. Double for 8 × 9 = 72
4. Double for 16 × 9 = 144

2. Halving Strategy (Table 5 and 15)

Leveraging the relationship between tables 10 and 5, and between 30 and 15, reduces calculation complexity significantly.

2.1 Table of 5 (Half of 10)

• Method: Multiply number by 10, then halve the result
• Example: 5 × 17 = (10 × 17) ÷ 2 = 170 ÷ 2 = 85
• Pattern Recognition: Results always end in 0 or 5 (even number → 0; odd number → 5)

2.2 Table of 15 (Half of 30)

• Method: Multiply number by 30, then halve the result
• Example: 15 × 8 = (30 × 8) ÷ 2 = 240 ÷ 2 = 120
• Alternative: (10 × number) + (5 × number) using halving for 5
• Example: 15 × 7 = 70 + 35 = 105

3. Tens-Based Anchoring (Tables 10, 20)

These tables serve as mental anchors for deriving nearby tables through addition or subtraction.

3.1 Table of 10 (Base Anchor)

• Rule: Append 0 to the multiplier (10 × 13 = 130)
• Use as Bridge: For table 9, calculate (10 × n) - n
• Use as Bridge: For table 11, calculate (10 × n) + n

3.2 Table of 20 (Double of 10)

• Method 1: Double the tens result (20 × 6 = 2 × 60 = 120)
• Method 2: Multiply by 2, then append 0 (20 × 6 = 12 with 0 = 120)

4. Neighbor Subtraction Method (Table 9)

Table 9 exhibits a unique digit sum property where product digits always sum to 9 (up to 9 × 10).

4.1 Primary Shortcut

• Formula: 9 × n = (10 × n) - n
• Example: 9 × 7 = 70 - 7 = 63
• Example: 9 × 14 = 140 - 14 = 126

4.2 Finger Method (Up to 9 × 10)

1. Number both hands' fingers 1 to 10 from left to right
2. For 9 × n, fold down the nth finger
3. Fingers left of folded finger = tens digit
4. Fingers right of folded finger = units digit
5. Example: 9 × 6: Fold 6th finger → 5 fingers left, 4 right → 54

4.3 Digit Sum Verification

• Property: For 9 × n (where n ≤ 10), digit sum of product = 9
• Example: 9 × 8 = 72 → 7 + 2 = 9 ✓
• Limitation: Pattern breaks after 9 × 10 (9 × 11 = 99 → 9 + 9 = 18)

5. Neighbor Addition Method (Table 11)

Table 11 demonstrates a repeating digit pattern for single-digit multipliers and systematic addition for larger numbers.

5.1 Repeating Digit Pattern (11 × 1 to 11 × 9)

• Rule: Write the multiplier twice side by side
• Examples: 11 × 3 = 33; 11 × 7 = 77; 11 × 9 = 99

5.2 Two-Digit Multipliers (Beyond 9)

• Method: 11 × n = (10 × n) + n
• Example: 11 × 13 = 130 + 13 = 143
• Visual Trick: For two-digit numbers like 11 × 25
• Write digits with space: 2_5
• Add the digits: 2 + 5 = 7
• Place sum in middle: 275 (works when sum <>

5.3 Carry-Over Cases

• Example: 11 × 67 using visual trick: 6_7 → 6 + 7 = 13
• Write 3 in middle, carry 1 to left: 6(+1)37 = 737

6. Decomposition Strategy (Table 12)

Table 12 breaks down using distributive property with easily calculated components.

6.1 Split into 10 + 2

• Formula: 12 × n = (10 × n) + (2 × n)
• Example: 12 × 8 = 80 + 16 = 96
• Example: 12 × 15 = 150 + 30 = 180

6.2 Alternative: Triple of 4

• Method: 12 = 3 × 4, so 12 × n = 3 × (4 × n)
• Example: 12 × 7 = 3 × 28 = 84
• Use Case: When 4 × n is easier to triple than adding 10n + 2n

7. Teen Tables (13, 14, 15, 16, 17, 18, 19)

These tables leverage base 10 addition combined with single-digit multiplication.

7.1 Universal Formula

• General Pattern: Teen × n = (10 × n) + (units digit × n)
• Example for 13: 13 × 6 = 60 + 18 = 78
• Example for 17: 17 × 5 = 50 + 35 = 85

7.2 Specific Shortcuts

7.2.1 Table of 14 (Double of 7)

• Method: Calculate 7 × n, then double the result
• Example: 14 × 9 = 2 × (7 × 9) = 2 × 63 = 126

7.2.2 Table of 15 (Covered in Section 2.2)

• Quick Recap: (10 × n) + (5 × n) or half of (30 × n)

7.2.3 Table of 16 (Covered in Section 1.1)

• Quick Recap: Quadruple doubling from 2

7.2.4 Table of 18 (Double of 9)

• Method: Calculate 9 × n, then double the result
• Example: 18 × 7 = 2 × 63 = 126
• Alternative: 18 × n = (20 × n) - (2 × n)
• Example: 18 × 8 = 160 - 16 = 144

8. Complementary Subtraction (High-Value Tables)

For tables close to 20, subtraction from 20 simplifies mental calculation.

8.1 Table of 19

• Formula: 19 × n = (20 × n) - n
• Example: 19 × 6 = 120 - 6 = 114
• Example: 19 × 12 = 240 - 12 = 228

8.2 Table of 18 (Alternative Method)

• Formula: 18 × n = (20 × n) - (2 × n)
• Example: 18 × 9 = 180 - 18 = 162

9. Middle-Range Tables (6, 7, 8)

These require hybrid approaches combining known tables and small adjustments.

9.1 Table of 6 (Double of 3)

• Method: Calculate 3 × n, then double it
• Example: 6 × 8 = 2 × 24 = 48
• Alternative: (5 × n) + n
• Example: 6 × 9 = 45 + 9 = 54

9.2 Table of 7 (Using 5 + 2 Split)

• Method: 7 × n = (5 × n) + (2 × n)
• Example: 7 × 8 = 40 + 16 = 56
• Memorization Priority: Table 7 has fewest shortcuts; focus on direct memorization

9.3 Table of 8 (Covered in Section 1.1)

• Quick Recap: Triple doubling from 2 (or double of 4)

10. Pattern Recognition Systems

Cross-table patterns enable verification and rapid mental checks during exams.

10.1 Units Digit Cycle Pattern

• Table 2: Units digits cycle: 2, 4, 6, 8, 0 (repeats every 5)
• Table 3: Units digits cycle: 3, 6, 9, 2, 5, 8, 1, 4, 7, 0 (repeats every 10)
• Table 4: Units digits cycle: 4, 8, 2, 6, 0 (repeats every 5)
• Table 5: Alternates 5, 0, 5, 0...
• Table 6: Units digits cycle: 6, 2, 8, 4, 0 (repeats every 5)
• Table 9: Units digits descend: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 (then repeats)

10.2 Divisibility Quick Checks

• All even tables (2, 4, 6, 8...): Products always even
• Table 5: Products end in 0 or 5
• Table 9: Sum of product digits divisible by 9 (up to reasonable limits)
• Table 3: Sum of product digits divisible by 3

10.3 Symmetry in Results

• Commutative Property: 7 × 8 = 8 × 7 = 56 (learn once, use twice)
• Strategic Memorization: For tables 2-10, only memorize combinations where first number ≤ second number
• Example: Know 3 × 9 = 27, automatically know 9 × 3 = 27

11. Advanced Calculation Techniques

For complex multipliers beyond 20, systematic decomposition and reconstruction maintains accuracy.

11.1 Distributive Property Mastery

• General Formula: a × (b + c) = (a × b) + (a × c)
• Example: 17 × 23 = 17 × (20 + 3) = (17 × 20) + (17 × 3) = 340 + 51 = 391
• Choose Split Wisely: Use multiples of 10, 5, or doubles for easier sub-calculations

11.2 Near-Multiple Strategy

• Method: For n × (m ± k), calculate n × m, then adjust by n × k
• Example: 8 × 19 = 8 × (20 - 1) = 160 - 8 = 152
• Example: 12 × 21 = 12 × (20 + 1) = 240 + 12 = 252

11.3 Cross-Multiplication for Adjacent Numbers

• For numbers like 15 × 15, 16 × 14, etc.:
• Take base (15): 15² = (15 × 10) + 5² = 150 + 25 = 225
• For 16 × 14: Base 15, difference ±1: (15²) - (1²) = 225 - 1 = 224
• Formula: (n - a)(n + a) = n² - a²

12. Strategic Memorization Framework

Systematic prioritization reduces total memorization load by 60-70% when shortcuts are applied correctly.

12.1 Priority Tiers

12.1.1 Tier 1: Direct Memorization Required

• Table 3: No consistent shortcut; memorize 3 × 1 through 3 × 10
• Table 7: Limited shortcuts; memorize 7 × 1 through 7 × 10
• Products 6 × 6 to 9 × 9: Frequent exam questions; must be instant recall

12.1.2 Tier 2: Shortcut-Derivable (Practice Until Automatic)

• Tables 9, 11, 12, 15, 18, 19: Strong patterns; use shortcuts initially, then memorize through repetition
• Target: Shortcut → Answer time under 2 seconds

12.1.3 Tier 3: Always Calculate (No Memorization Needed)

• Tables 2, 4, 5, 8, 10, 16, 20: Doubling/halving/appending patterns so fast they need no memorization
• Tables 13, 14, 17: Rapid calculation via 10n + additions

12.2 Spaced Repetition Protocol

1. Day 1-3: Focus on Tier 1 tables with 5 repetitions daily
2. Day 4-7: Add Tier 2, practice shortcuts 3 times daily until automatic
3. Day 8-14: Mixed practice all tables, random order, 50 questions daily
4. Week 3 onward: Maintenance practice 20 questions daily, focus on weak areas

13. Common Mistakes and Traps

Identifying systematic errors prevents recurring mistakes in high-pressure exam scenarios.

13.1 Calculation Errors

• Doubling Cascade Errors: When using 2→4→8→16 chain, losing track after second doubling; write intermediate steps until automatic
• Carry-Over Mistakes in 11 × n: For 11 × 78, forgetting to carry in visual trick: 7_8 → 7 + 8 = 15 → correct answer 858, not 7158
• Sign Errors in Subtraction Method: Calculating 19 × 7 as (20 × 7) + 7 instead of (20 × 7) - 7

13.2 Pattern Misapplication

• 9's Digit Sum Beyond Limit: Applying digit-sum-equals-9 rule to 9 × 15; rule works reliably only up to 9 × 10
• 15 Halving with Odd Results: Calculating 15 × 9 as half of 30 × 9 = 270 ÷ 2 = 135 ✓ (correct), but making arithmetic errors in halving odd hundreds
• 11 Repeating Pattern Overgeneralization: Applying double-digit rule to 11 × 13 expecting 1133; rule applies only to 11 × 1 through 11 × 9

13.3 Speed vs Accuracy Trade-off

• Trap: Rushing through 7 × 8 or 6 × 7 leads to confusion with nearby products
• Solution: For Tier 1 products, always verify using nearest table (7 × 8 = 56 is between 7 × 7 = 49 and 7 × 9 = 63)
• Confusing 6 × 7 = 42 with 6 × 8 = 48: Remember "42 comes before 48" using alphabetical order (six-seven before six-eight)

14. Exam-Day Application Strategies

Contextual deployment of shortcuts optimizes time management during actual examination conditions.

14.1 Method Selection Protocol

• If multiplier ≤ 10: Use direct recall or fastest single shortcut
• If multiplier 11-20: Use complementary addition (11-12) or subtraction (18-20)
• If multiplier > 20: Decompose into (10a + b) form, calculate separately

14.2 Scratch Work Optimization

• Vertical Alignment: Write intermediate steps vertically to prevent digit displacement errors
• Example for 17 × 14:

170 (10 × 17) + 98 (4 × 17 using double of 7 × 17 = 119 ÷ 2... no, 4 × 17 = 68)

• Correction: 4 × 17: double 2 × 17 = 34 to get 68
• Final: 170 + 68 = 238

14.3 Quick Verification Methods

• Divisibility Check: Ensure product is divisible by multiplicand and multiplier
• Digit Sum Check: For table 3 or 9, verify digit sum divisibility
• Reasonableness Check: Estimate using nearest 10s: 17 × 14 should be near 17 × 15 = 255 (our answer 238 is close ✓)
• Last Digit Verification: Check units digit using cycle patterns from Section 10.1

Mastery of these shortcut methods transforms table calculations from time-consuming memorization into pattern-recognition exercises. Prioritize Tier 1 memorization while practicing Tier 2 shortcuts until they become automatic reflexes. Regular mixed practice under timed conditions embeds these patterns into procedural memory, enabling rapid, accurate calculations essential for competitive exam success. The key principle is not memorizing 19 separate tables, but understanding 6-7 core patterns that generate all required products efficiently.