Times Tables: 6 Through 10

For Elementary Students

You Already Know Half of These!

Great news: If you learned the 1–5 times tables, you already know HALF of the 6–10 tables!

Think about it like this: Multiplication works both ways!

3 × 6 = 18
6 × 3 = 18

Same answer! Just flipped!

So when you see 6 × 3, you can think "That's the same as 3 × 6, and I already know that's 18!"

The 6 Times Table

Let me show you the new facts you need to learn:

6 × 6 = 36
6 × 7 = 42
6 × 8 = 48
6 × 9 = 54
6 × 10 = 60

Cool Pattern for 6s: When you multiply 6 by an even number, the answer ends in that same digit!

6 × 2 = 12  (ends in 2!)
6 × 4 = 24  (ends in 4!)
6 × 6 = 36  (ends in 6!)
6 × 8 = 48  (ends in 8!)
6 × 10 = 60 (ends in 0!)

The 7 Times Table

The 7s are tricky—no magic pattern here. Just practice!

7 × 7 = 49
7 × 8 = 56
7 × 9 = 63
7 × 10 = 70

Memory trick: "5, 6, 7, 8 → 56 = 7 × 8"

Say it like a rhythm: "Five-six-seven-eight makes fifty-six!"

The 8 Times Table

8 × 8 = 64
8 × 9 = 72
8 × 10 = 80

The Double-Double-Double Trick!

Since 8 = 2 × 2 × 2, you can double three times!

Example: What's 8 × 7?

Start with 7
Double it: 7 × 2 = 14
Double again: 14 × 2 = 28
Double again: 28 × 2 = 56

Answer: 8 × 7 = 56!

Or think: "8 is double of 4, and I know 4 × 7 = 28, so 8 × 7 = 56!"

The 9 Times Table (THE COOLEST!)

The 9s table has AMAZING patterns!

Pattern 1: Digits add up to 9!

9 × 1 = 09  →  0 + 9 = 9 ✓
9 × 2 = 18  →  1 + 8 = 9 ✓
9 × 3 = 27  →  2 + 7 = 9 ✓
9 × 4 = 36  →  3 + 6 = 9 ✓
9 × 5 = 45  →  4 + 5 = 9 ✓
9 × 6 = 54  →  5 + 4 = 9 ✓
9 × 7 = 63  →  6 + 3 = 9 ✓
9 × 8 = 72  →  7 + 2 = 9 ✓
9 × 9 = 81  →  8 + 1 = 9 ✓
9 × 10 = 90 →  9 + 0 = 9 ✓

Pattern 2: The tens digit goes UP, the ones digit goes DOWN!

9 × 1 = 09  (0 and 9)
9 × 2 = 18  (1 and 8)
9 × 3 = 27  (2 and 7)
9 × 4 = 36  (3 and 6)
9 × 5 = 45  (4 and 5)

See? Tens go: 0, 1, 2, 3, 4...
Ones go: 9, 8, 7, 6, 5...

The FINGER TRICK for 9s!

This is like magic! Hold up all 10 fingers.

To find 9 × 4:

1. Number your fingers 1–10 (left to right)
2. Put down finger #4
3. Look at your hands:

[1][2][3] DOWN [5][6][7][8][9][10]
   3 fingers  |  6 fingers

Answer: 3 and 6 → 36!

To find 9 × 7:

Put down finger #7:

[1][2][3][4][5][6] DOWN [8][9][10]
      6 fingers     |    3 fingers

Answer: 6 and 3 → 63!

The 10 Times Table

This is the EASIEST of all! Just add a zero!

10 × 1 = 10
10 × 2 = 20
10 × 3 = 30
10 × 4 = 40
10 × 7 = 70
10 × 12 = 120
10 × 99 = 990

Rule: Whatever number you multiply by 10, just stick a zero at the end!

Quick Review Chart

    6s    7s    8s    9s   10s
  ----  ----  ----  ----  ----
×6  36    42    48    54    60
×7  42    49    56    63    70
×8  48    56    64    72    80
×9  54    63    72    81    90
×10 60    70    80    90   100

Memory Tricks

For 6s: "Even endings match!"

For 7s: "Five-six-seven-eight makes fifty-six!" (7 × 8 = 56)

For 8s: "Double the 4s table!"

For 9s: "Finger trick for the win!"

For 10s: "Just add zero!"

Quick Tips

Tip 1: Use what you already know! (3 × 7 = 21, so 7 × 3 = 21)

Tip 2: Practice the hard ones (7s and 8s) more!

Tip 3: The 9s finger trick works every time!

Tip 4: If you forget one, use a trick (like doubling for 8s)

For Junior High Students

Understanding the 6–10 Multiplication Tables

The times tables from 6 through 10 represent multiplication facts that extend beyond the foundational 1–5 range. Understanding these requires recognizing patterns, applying the commutative property, and building on previously learned facts.

Definition: The n times table consists of the products n × k for all integers k in a given range (typically 1–10 or 1–12).

Commutative property: For any integers a and b, a × b = b × a

Implication: Learning 6 × 7 = 42 automatically gives you 7 × 6 = 42, reducing the number of facts to memorize.

Strategic Learning Approach

New facts to learn: When you already know tables 1–5, you only need to learn the "upper diagonal" of the multiplication table:

      6    7    8    9   10
    ---- ---- ---- ---- ----
6  | 36   42   48   54   60
7  |      49   56   63   70
8  |           64   72   80
9  |                81   90
10 |                    100

Total new facts: 15 (compared to 25 if learning all 6–10 combinations)

Efficiency gain: 40% reduction in memorization required

The 6 Times Table

Products:

6 × 6 = 36
6 × 7 = 42
6 × 8 = 48
6 × 9 = 54
6 × 10 = 60

Pattern: Even number property

When multiplying 6 by an even number 2k, the result ends in the same digit as k.

Proof: 6 × (2k) = 12k

The units digit of 12k depends on the units digit of k:

• If k ends in 1: 12 × 1 = 12 (ends in 2)
• If k ends in 2: 12 × 2 = 24 (ends in 4)
• If k ends in 3: 12 × 3 = 36 (ends in 6)
• Pattern continues...

The 7 Times Table

Products:

7 × 7 = 49
7 × 8 = 56
7 × 9 = 63
7 × 10 = 70

Characteristics: No simple digit pattern exists for the 7 times table. This is because 7 is coprime to 10 (gcd(7, 10) = 1), so the units digits cycle through all non-zero values.

Cycle: 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, then repeat

Strategy: Focus on memorization and repeated practice. Use relationships to known facts (e.g., 7 × 8 = 7 × 7 + 7 = 49 + 7 = 56).

The 8 Times Table

Products:

8 × 8 = 64
8 × 9 = 72
8 × 10 = 80

Relationship to powers of 2: Since 8 = 2³, multiplying by 8 is equivalent to tripling the doubling operation.

Algebraic relationship: 8 × n = 2 × (4 × n)

Application: If you know the 4 times table, double those products.

Example: 4 × 9 = 36, so 8 × 9 = 2 × 36 = 72

Alternative: 8 × n = (10 × n) - (2 × n)

Example: 8 × 7 = 70 - 14 = 56

The 9 Times Table

Products:

9 × 1 = 09
9 × 2 = 18
9 × 3 = 27
9 × 4 = 36
9 × 5 = 45
9 × 6 = 54
9 × 7 = 63
9 × 8 = 72
9 × 9 = 81
9 × 10 = 90

Pattern 1: Digital root property

For any product 9 × n (where 1 ≤ n ≤ 10), the sum of digits equals 9.

Mathematical basis: 9 × n = 10n - n

In base 10, this creates products where digit sums equal 9.

Pattern 2: Tens and units relationship

For 9 × n (where 1 ≤ n ≤ 10):

• Tens digit = n - 1
• Units digit = 10 - n
• Check: (n - 1) + (10 - n) = 9 ✓

Example: 9 × 7

• Tens digit: 7 - 1 = 6
• Units digit: 10 - 7 = 3
• Result: 63 ✓

Algebraic explanation:

9 × n = 10n - n
     = 10(n - 1) + (10 - n)

This shows why the tens digit is (n - 1) and units digit is (10 - n).

The finger algorithm:

A physical mnemonic device that encodes the tens-units pattern:

1. Assign fingers 1–10 (left to right)
2. For 9 × n, fold down finger n
3. Fingers left of folded finger = tens digit
4. Fingers right of folded finger = units digit

Why it works: Folding finger n leaves (n - 1) fingers to the left and (10 - n) to the right, matching the algebraic formula.

The 10 Times Table

Products:

10 × 1 = 10
10 × 2 = 20
10 × 3 = 30
...
10 × n = 10n

Property: Multiplying by 10 in base 10 shifts all digits one place to the left, equivalent to appending a zero.

Algebraic representation: 10 × n = n × 10¹

Decimal shift: In positional notation, this increases each digit's place value by a factor of 10.

Example: 10 × 47 = 470

• The 4 shifts from tens to hundreds place
• The 7 shifts from ones to tens place
• A new 0 fills the ones place

Generalization: Multiplying by 10^k appends k zeros.

Applications

Algebraic manipulation:

Knowing these facts enables faster mental math in algebra.

Example: Expand 7(x + 8)

= 7x + 7 × 8
= 7x + 56

Instant recall of 7 × 8 = 56 accelerates the process.

Area calculations:

Example: Rectangle with dimensions 9 m × 7 m

Area = 9 × 7 = 63 m²

Percentage calculations:

10% of any number is that number divided by 10 (or multiplied by 0.1).

Example: 10% of 85 = 8.5

Scaling and proportions:

Example: Recipe calls for 6 eggs for 8 people. How many for 7 people?

Understanding 6 × 7 and 6 × 8 helps set up the proportion efficiently.

Memorization Strategies

Chunking: Group facts by patterns (e.g., all 9s products have digit sum 9)

Distributed practice: Review facts regularly over time rather than cramming

Interleaving: Mix practice of different tables rather than practicing one table repeatedly

Retrieval practice: Test yourself frequently rather than just re-reading

Elaboration: Connect new facts to known facts (8 × 7 = double of 4 × 7)

Common Errors

Error 1: Confusing similar products

• 7 × 8 = 56 (not 54 or 63)
• 6 × 9 = 54 (not 56 or 63)

Strategy: Create distinct associations for each fact

Error 2: Applying pattern incorrectly

❌ Assuming all tables have simple digit patterns ✓ Recognizing that 7 has no simple pattern

Error 3: Calculation mistakes with strategies

When using "10n - n" for 9s table: ❌ 9 × 8 = 80 - 8 = 78 ✓ 9 × 8 = 80 - 8 = 72

Extensions

Modular arithmetic: Times tables modulo 10 reveal patterns in units digits.

Example: Units digits of 7 times table form the sequence 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, which is a permutation of all digits.

Matrix representation: Multiplication tables can be represented as matrices, revealing symmetry (commutative property) along the diagonal.

Number theory: Products reveal divisibility rules and prime factorizations.

Example: 72 = 8 × 9 = 2³ × 3²

Summary

Mastery approach:

1. Use commutative property to reduce memorization
2. Apply patterns where available
3. Build on known facts (4s → 8s)
4. Practice retrieval regularly
5. Verify using digit patterns (especially for 9s)

Practice