Math Shortcuts for Middle Schoolers

1. Multiplying by 5 Quickly

Description: Multiplying by 5 is often easier when you think in terms of division and multiplication. First, halve the number (divide by 2), then multiply the result by 10.
Example:
- 46 × 5: 46 ÷ 2 = 23, 23 × 10 = 230.
- 72 × 5: 72 ÷ 2 = 36, 36 × 10 = 360.

2. Square Numbers Ending in 5

Description: Numbers ending in 5 have a pattern when squared. Multiply the first digit by the next higher digit, then append 25.
Example:
- 25²: 2 × 3 = 6, append 25 → 625.
- 65²: 6 × 7 = 42, append 25 → 4225.

3. The 11 Multiplication Trick

Description: When multiplying a two-digit number by 11, add the digits together and place the sum between the digits.
Example:
- 52 × 11: 5 + 2 = 7, place 7 between 5 and 2: 572.
- 81 × 11: 8 + 1 = 9, place 9 between 8 and 1: 891.
Note: If the sum is greater than 9, carry over the extra.
- 78 × 11: 7 + 8 = 15, place 5 between, carry 1 to the 7 → 858.

4. Divisibility Rules

Description: Simple rules help determine if a number is divisible without full division.
- Divisible by 2: Last digit is 0, 2, 4, 6, or 8.
- Divisible by 3: Sum of digits is divisible by 3.
  Example: 123 → 1 + 2 + 3 = 6, divisible by 3.
- Divisible by 9: Sum of digits is divisible by 9.
  Example: 729 → 7 + 2 + 9 = 18, divisible by 9.
- Divisible by 5: Last digit is 0 or 5.

5. Finding Percentages Easily

Description: Break percentages into smaller, manageable parts.
Example:
- Find 35% of 200: 10% = 20, 30% = 20 × 3 = 60, 5% = 10, Total: 60 + 10 = 70.
- Find 18% of 50: 10% = 5, 8% = 4, Total = 9.

6. Multiplying by 9 (Finger Trick)

Description: For 9 × n (where n is a number from 1 to 10), fold the n-th finger. Count fingers to the left for the tens digit, and fingers to the right for the units digit.
Example:
- 9 × 4: Fold the 4th finger. Left = 3, Right = 6 → 36.
- 9 × 7: Fold the 7th finger. Left = 6, Right = 3 → 63.

7. Subtracting from 1000

Description: For subtraction like 1000 - n, subtract each digit of n from 9, except the last digit from 10.
Example:
- 1000 - 576: 9-5 = 4, 9-7 = 2, 10-6 = 4 → 424.
- 1000 - 89: Add leading zero → 089: 9-0 = 9, 9-8 = 1, 10-9 = 1 → 911.

8. Multiplying Numbers Close to 100

Description: Subtract each number from 100, subtract the results from 100, and multiply the differences.
Example:
- 97 × 96: 100-97 = 3, 100-96 = 4, 100-(3+4) = 93, 3 × 4 = 12, Answer = 9312.
- 98 × 99: 100-98 = 2, 100-99 = 1, 100-(2+1) = 97, 2 × 1 = 2, Answer = 9702.

9. Average of Consecutive Numbers

Description: For consecutive numbers, the average is the middle number. If even count, it’s the average of the middle two numbers.
Example:
- 5, 6, 7, 8, 9: Middle is 7.
- 4, 5, 6, 7: Middle two are 5 and 6, average = (5+6)/2 = 5.5.

10. Quick Addition of Large Numbers

Description: Break numbers into smaller parts (hundreds, tens, units), add separately, and combine.
Example:
- 394 + 576: Hundreds: 300 + 500 = 800, Tens: 90 + 70 = 160, Units: 4 + 6 = 10, Total: 800 + 160 + 10 = 970.
- 738 + 246: Hundreds: 700 + 200 = 900, Tens: 30 + 40 = 70, Units: 8 + 6 = 14, Total: 900 + 70 + 14 = 984.