Advance A

W1·D4

Special Multiplication Patterns

Squares ending in 5, and the same-first-digit / sum-to-10 trick.

Learning Objectives

  • 01

    Square any number ending in 5 in one second.

  • 02

    Multiply two numbers with the same first digit when the last digits sum to 10.

Topics

Squaring numbers ending in 5Same first digits, last digits sum to 10

Worked Examples

35²=1225

Squares-ending-in-5 shortcut (Benjamin, Secrets of Mental Math).

  1. 01

    3 × 4 = 12

    Drop the 5. Multiply the front digit(s) by the next number: 3 × 4 = 12.

  2. 02

    12 | 25 → 1225

    Stick "25" on the end. Every square ending in 5 ends in 25 — because (10a+5)² = 100·a(a+1) + 25.

  3. →

    = 1225

105²=11025

Squares-ending-in-5 shortcut (Benjamin, Secrets of Mental Math).

  1. 01

    10 × 11 = 110

    Drop the 5. Multiply the front digit(s) by the next number: 10 × 11 = 110.

  2. 02

    110 | 25 → 11025

    Stick "25" on the end. Every square ending in 5 ends in 25 — because (10a+5)² = 100·a(a+1) + 25.

  3. →

    = 11025

84 × 86=7224

Same-first-digit / units-sum-to-10 shortcut (Benjamin).

  1. 01

    8 × 9 = 72

    Front digits are the same (8). Multiply 8 by 9 → 72.

  2. 02

    4 × 6 = 24

    Units sum to 10 (4 + 6 = 10), so just multiply them: 4 × 6 = 24.

  3. 03

    72 | 24 → 7224

    Glue the two pieces together (pad the units product to 2 digits).

  4. →

    = 7224

62 × 68=4216

Same-first-digit / units-sum-to-10 shortcut (Benjamin).

  1. 01

    6 × 7 = 42

    Front digits are the same (6). Multiply 6 by 7 → 42.

  2. 02

    2 × 8 = 16

    Units sum to 10 (2 + 8 = 10), so just multiply them: 2 × 8 = 16.

  3. 03

    42 | 16 → 4216

    Glue the two pieces together (pad the units product to 2 digits).

  4. →

    = 4216

68 × 62=4216

Same-first-digit / units-sum-to-10 shortcut (Benjamin).

  1. 01

    6 × 7 = 42

    Front digits are the same (6). Multiply 6 by 7 → 42.

  2. 02

    8 × 2 = 16

    Units sum to 10 (8 + 2 = 10), so just multiply them: 8 × 2 = 16.

  3. 03

    42 | 16 → 4216

    Glue the two pieces together (pad the units product to 2 digits).

  4. →

    = 4216

Benjamin, Secrets of Mental Math — same-tens, units-sum-10 shortcut

Talking Points

  • •Why it works: (10a+5)² = 100·a(a+1) + 25.
  • •Same-first-digit trick requires last digits to sum to exactly 10.

Week 1 · Fundamentals

Mental Math Fundamentals & Basic Operations

  • →Develop left-to-right calculation habits
  • →Strengthen multiplication table mastery
  • →Learn fundamental mental shortcuts

Sample Problems · Week 1

One simple and one intermediate problem to preview the week.

Simple · Day 1

57 + 38=95

Intermediate · Day 3

314 × 11=3454