Multiplication Facts: 7, 8, and 9 - Third Grade Mathematics

The 7s, 8s, and 9s are often viewed as the most challenging multiplication facts, but they become completely approachable when you unlock their underlying patterns. In this chapter, you will learn the "5 + 2" strategy for sevens, the "triple double" strategy for eights, and the famous finger tricks and tens patterns for nines.

The Sevens Family: The 5 + 2 Strategy

Because 7 can be split into 5 + 2, you can break any 7 fact into a friendly 5 fact plus a friendly 2 fact!

Formula: 7 x n = (5 x n) + (2 x n)

Let us solve 7 x 6:
- 5 x 6 = 30

- 2 x 6 = 12

- 30 + 12 = 42
Therefore: 7 x 6 = 42!

Let us solve 7 x 8:
- 5 x 8 = 40

- 2 x 8 = 16

- 40 + 16 = 56
Therefore: 7 x 8 = 56! A fun way to remember 7 x 8 = 56 is the counting sequence: 5, 6, 7, 8 -> 56 = 7 x 8!

7 x 1 = 7
7 x 2 = 14
7 x 3 = 21
7 x 4 = 28
7 x 5 = 35
7 x 6 = 42
7 x 7 = 49
7 x 8 = 56
7 x 9 = 63
7 x 10 = 70

The Eights Family: Double, Double, Double!

Since 8 is 2 x 2 x 2, multiplying by 8 is simply doubling three times!

Strategy: Double three times.

Example: 8 x 7
- First double: 7 + 7 = 14
- Second double: 14 + 14 = 28
- Third double: 28 + 28 = 56
Therefore: 8 x 7 = 56!

Example: 8 x 6
- Double 1: 6 + 6 = 12
- Double 2: 12 + 12 = 24
- Double 3: 24 + 24 = 48
Therefore: 8 x 6 = 48!

8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80

The Nines Family: The 10 - 1 Strategy and Digit Patterns

The 9s family has some of the most fascinating patterns in all of mathematics!

Strategy 1: Multiply by 10, Then Subtract One Group

Because 9 is 10 - 1, you can multiply by 10 and then subtract the number:
- 9 x 6 = (10 x 6) - 6 = 60 - 6 = 54

- 9 x 8 = (10 x 8) - 8 = 80 - 8 = 72

- 9 x 4 = (10 x 4) - 4 = 40 - 4 = 36


Strategy 2: The Sum of Digits Always Equals 9!

Look closely at the 9s products:

9 x 1 = 09  ->  0 + 9 = 9
9 x 2 = 18  ->  1 + 8 = 9
9 x 3 = 27  ->  2 + 7 = 9
9 x 4 = 36  ->  3 + 6 = 9
9 x 5 = 45  ->  4 + 5 = 9
9 x 6 = 54  ->  5 + 4 = 9
9 x 7 = 63  ->  6 + 3 = 9
9 x 8 = 72  ->  7 + 2 = 9
9 x 9 = 81  ->  8 + 1 = 9
9 x 10 = 90 ->  9 + 0 = 9

Notice two amazing things:
1. The tens digit is always 1 less than the factor you are multiplying by (for 9 x 7, the tens digit is 6).

2. The ones digit is whatever number is needed to make the digits sum to 9 (since 6 + 3 = 9, the product is 63)!


Chapter Practice Exercises

  1. Calculate the products: a. 7 x 7 b. 8 x 8 c. 9 x 9 d. 7 x 9 e. 8 x 9
  2. Solve 7 x 4 using the (5 + 2) strategy. Show the two friendly facts and the final sum.
  3. Solve 8 x 9 using the "triple double" strategy on 9.
  4. Use the 10 - 1 strategy to calculate 9 x 7.
  5. An octopus has 8 tentacles. How many tentacles do 7 octopuses have in total?

Solutions and Step-by-Step Answers

  1. Products: a. 7 x 7 = 49. b. 8 x 8 = 64. c. 9 x 9 = 81. d. 7 x 9 = 63. e. 8 x 9 = 72.
  2. For 7 x 4:
  3. 5 x 4 = 20
  4. 2 x 4 = 8
  5. 20 + 8 = 28.
  6. For 8 x 9:
  7. Double 1: 9 + 9 = 18
  8. Double 2: 18 + 18 = 36
  9. Double 3: 36 + 36 = 72 Answer: 8 x 9 = 72.
  10. For 9 x 7:
  11. 10 x 7 = 70
  12. Subtract 7: 70 - 7 = 63.
  13. 7 octopuses with 8 tentacles each: 7 x 8 = 56 tentacles in total.