The logical way to learn multiplication facts

Once upon a time, children learned multiplication facts by reciting them in order. “…three times five is fifteen, three times six is eighteen, three times seven is twenty-one…” This rote method doesn’t require any understanding, and you remember the information as a group. Especially at the beginning, a student may have to go through the whole 7s row to find the answer to 7 x 8. That’s really inefficient, and many students don’t even notice the similarity between 3 x 4 and 4 x 3.

Learning math facts in random order (by worksheets or flash cards) is an improvement, but still doesn’t require any understanding. A child may know that 3 x 2 = 6, but will that help her when she can’t remember 3 x 4?

Math is unfailingly logical, from 2 + 2 all the way up to the derivatives of trigonometric functions. Teach it logically from the beginning, and students will have a much better foundation for understanding all the rules and processes in the entire structure. There’s a logical way to learn math facts that works faster and promotes deeper understanding of how numbers work.

Refer to my color-coded multiplication table while reading.

The multiplication table from 0s through 12s includes 169 math facts. Daunting, isn’t it? But right off the bat, we can eliminate almost half of them, because 3 x 4 and 4 x 3 are the same. Order doesn’t matter for multiplication. I’ve colored the duplicates gray, so now we’re down to 91 math facts.

The easy ones: The orange facts are labeled “easy”, because there are very simple rules for them. Multiply anything by 0, and you get 0. Multiply anything by 1, and you get the same thing you started with. Multiplying by 2 is the same as adding a number to itself, so if you know addition facts, you know all the 2s. Multiply anything by 10, and you just add a 0 to what you already had. Multiply any one digit by 11, and you get two of that digit side-by-side.

53 more facts knocked out – we’re cruising right along! Now there are only 38 multiplication facts left, and many more shortcuts. There are multiple tricks each for 5s and 9s, a trick for sixes, a trick for squares, and a trick for the bigger 11s.

5s: Multiplication is repeated addition, so one way to multiply is to count by the multiplier, as many times as necessary. To multiply 6 x 5, count by 5s six times: 5, 10, 15, 20, 25, 30. Another way is to divide by 2, then put a 0 after it if even, or a 5 if odd. For 6 x 5, divide 6 in half to get 3. Now put a 0 after it and you have 30. For 7 x 5, 7 is odd – dividing in half doesn’t work out evenly. So divide 6 in half to get 3, then put a 5 after it instead of a 0, and you have 35. Cool!

9s: All the multiples of 9 have digits that add up to 9. Multiplying by 9 is less than multiplying by 10, so one way to do nines is to multiply one less by 10, then add whatever digit make the digits equal 9. To use this trick for 4 x 9, you would start with one less than 4, which is 3, and multiply by 10. 3 + 6 = 9, so the answer is 36. Another way is to use the finger trick. Hold out your hands in front of you, fingers spread. To multiply 4 x 9, put down the 4th finger, counting from the left. Now there are 3 fingers to the left side, and 6 fingers to the right side of the bent finger.

That’s 15 more we’ve covered. Only 23 left! Don’t you feel smart?

This would be a good time to start practicing what you’ve learned so far. At Worksheet Works you can make a worksheet that includes only certain fact families. Set the range from 0 to 11, choose 0, 1, 2, 5, 9, 10, 11, and click “Randomly reverse…” so you get used to seeing the facts both ways. When teaching multiplication facts to a student for the first time, they’ll need to start practicing after the 2s, 11s, or 5s.

6s: I only recently learned this trick for 6s. If you want six multiplied by an even number, put half the number in the tens place, and the number itself in the ones place. 6 x 4 is 2 (half of 4) followed by the 4. But what about the odds? We already covered 6 x 5 and 6 x 9. 6 x 3 and 6 x 7 are best handled by doubling or by adding on, which I’ll get to in a minute.

Squares:A square means a number times itself. We already know how to do 5 x 5, 6 x 6, and 9 x 9, from those families. But there’s a trick to get any square. Look at 5 x 5 = 25 in the table, and notice that 4 x 6, just one down and to the left, is one less. This is true for all the squares. For any square, multiply the next bigger number by the next smaller number, and add 1. So to find 3 x 3, you can use 2 x 4 = 8 and add 1 to get 9. Try it for 4 x 4, 7 x 7, and 8 x 8. This pattern is included on the second page of the multiplication chart.

Bigger 11s: One more trick for 11s. You know how to multiply a one-digit number by 11. When you multiply any two-digit number by 11, you get the same two digits, with a new one in between. The one in between is the two original digits added together. So 11 x 24 = 264 because 2 + 4 = 6 and the six goes in the middle. You can use this trick to do 11 x 11 and 11 x 12.

With the 6s, the squares, and the 11s, you’ve learned 9 more math facts, and there are only 14 left to learn. Keep practicing with the pages from Worksheet Works.

There are two general methods that apply to the math facts above, and are the only way to get the last few facts: adding on, and doubling.

Adding on: Multiplication is repeated addition, but you don’t necessarily have to start from zero. 3 x 4 is 3 added 4 times, or 3 + 3 + 3 + 3. But if you already know what 3 x 3 is, you don’t have to do that part; you can just add another 3 to get 3 x 4. If this is confusing, use some graph paper and color in 3 rows of 3 squares each for 3 x 3, then another row of 3 squares to make 3 x 4. Another column of 4 squares would make 4 x 4. Another row of 4 squares would make 5 x 4, and so on.

Doubling: Yet another method, faster than adding on, is to double a smaller fact. This pattern is also on the second page of the multiplication chart. If you want 3 x 4, 4 is 2 doubled, so you can use 3 x 2 and double that. 3 x 2 = 6, so 3 x 4 is 6 doubled: 12. Notice that if you doubled the 3 instead, 6 x 2, the answer would still be 12. If you double 3 x 4 or 6 x 2, you get 6 x 4 or 3 x 8, which are both 24. Again, if this is confusing, graph paper can help. A 3 x 2 rectangle, next to another 3 x 2 rectangle, is a 3 x 4 rectangle. Double that one, and you have a 3 x 8 rectangle. Doubling is the easiest way to do most of the blue facts, and 7 x 3 is the only one that requires adding on.

The squares & doubles page is a good “cheat sheet” for when you’re practicing math facts. When you can’t remember one of these, you can refer to the sheet to figure out which other fact will get you there. Keep practicing with worksheets, flash cards, and games, until you can see a multiplication fact and know the answer immediately. You’ll be amazed at how much your math skills improve!

Why memorize math facts?

You’ll do simple math without even thinking. You’ll be able to do bigger problems (like adding or multiplying several digits) on paper, and eventually in your head.

You’ll work faster and yet make fewer mistakes because math facts will be automatic.

You’ll be less dependent on the calculator, which will also lead to fewer mistakes. And when you don’t have a calculator handy, it won’t matter.

The more you practice, you’ll develop an intuition about math. You’ll notice relationships between numbers and understand concepts in a deeper way. For example, you’ll discover shortcuts that will allow you to do calculations in your head while grocery shopping or doing carpentry.

You’ll have greater confidence and become more independent, which will snowball into greater confidence.

You’ll be more able to focus on everything else in math — dealing with several digits, learning new concepts such as fractions, setting up word problems correctly, and thinking about whether your answers make sense.

Coming soon: Easy ways to learn math facts