Knowing your 3 times table is one thing; teaching it to someone else is another thing entirely. It's best to keep it simple, and repetition is often the best way to learn any multiplication table. But if that doesn't work, you have many other tricks at your disposal.
TL;DR (Too Long; Didn't Read)
Repetition, a multiplication grid, flash cards and memory games are the best ways to learn (and teach) the 3 times table. These are tricks for all multiplication tables, not just the 3 times table.
Create a Multiplication Grid
Create a grid of 121 squares (11 squares across and 11 squares down). Leave the top left square blank and write the numbers 1 to 10 in each square from left to right and top to bottom. Where the top row numbers and left hand side numbers intersect, write the product of both numbers (i.e., 3 × 2 = 6).
Fill the entire grid, but concentrate on the 3-times products. Use a highlighter pen to highlight the vertical columns and horizontal rows, beginning with the number 3, which both contain the numbers 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30. In other words, look for those numbers among the products that you wrote in the squares, and highlight the intersecting rows and columns. Point out any patterns in the 3 times table, such as the numbers have the sum of their digits coming to 3, then 6, then 9. For example, for 12, 1 + 2 = 3; for 15, 1 + 5 = 6; and for 18, 1 + 8 = 9.
Sciencing Video Vault
Because you've filled the entire grid, you can reuse it when you're ready to move on to other times tables, such as the 4 times table.
Use Flash Cards
Make flash cards with the problem on the front (e.g., 3 × 8) and the answer (24) on the back until you have a card for each number in the 3 times table. The act of writing the problems and answers is another form of repetition. Hold up the flash cards in different orders for the child to answer. Use a timer to give him targets to beat as he gets better at remembering the correct answers.
Play a Memory Game
A good memory game for the 3 times table uses separate problem cards and number cards. Create the problem cards in different shapes than the number cards (for example, rectangles and circles). Place all cards face-down on the table. Ask one player to turn over one of the problem cards and find the number card with the correct answer. If the cards match, she keeps them to one side; if not, she turns them over again and another player takes a turn. The winner is the player with the most cards at the end of the game.
Split the Table
For anyone struggling to learn the 3 times table, split it in two to make it a smaller, more manageable task. Practice up to 3 × 5 first, and then work on the rest of the table after the first part is mastered. Note any particularly difficult problems, such as 3 × 9, and focus on those when the other problems have been mastered.