Mystery Skill 1: 63 Times

Published on Thursday, August 08, 2013 in , , , , ,

SPUI's 63 highway shield imageI'm going to take an unusual approach for my next several posts. I'm going to help you develop an unusual mathematical skill, but I'm going to save the nature of that skill as s surprise.

I'll start today by teaching you a particular math shortcut you'll need to develop. On Sunday, I'll teach you a similar, related math shortcut. In Next Thursday's post, I'll show you how to put these two simple shortcuts together in a way that gives you an impressive and surprising skill.

The first thing you'll need to learn is how to multiply by 63 quickly.

The first thing that will make this feat easier is that you're going to write parts of it down as you go. This will take pressure of your memory, as well as make the whole process more visual for your audience.

Before learning this skill, you should already be comfortable multiplying numbers up to 33 by 3 in your head. You should also be able to quickly multiply numbers up to 200 by 2 in your head, as well.

WHOLE NUMBERS: Here's the basic process, which will seem abstract at first. Don't worry, the practical examples that follow should help clear up any confusion. We start with a given number from 1 to 33:

1) Multiply the given number by 3.
2) Write down the ones digit of this answer on paper.
3) Take the amount you calculated in step 1, and double that number. Do not forget the number from step 1, however.
4) Take the number from step 3, and add ONLY the tens digit of the number from step 1 to it.
5) Write the total from step 4 on the paper, to the immediate left of the digit written in step 2.

If the references to the ones place and tens place confuse you, here's a quick refresher on place value.

Let's try running through this process with 12 as the given number.

The first step is to mentally multiply 12 by 3, which is 36. For the second step, we simply write down the number 6 on the paper, because the ones digit of 36 is 6.

In step 3, we're going to mentally double 36 to get 72, so now we're thinking of 36 AND 72, as we're not supposed to forget the number calculated in step 1.

Step 4 tells us to add the number from step 3, and add it to the tens digit of the number from step 1. In our example, the tens digit of 36 is 3, so we add 72 (the number calculated in step 3) plus 3 (the tens digit of 36) to get 75.

The final step is easy, we write that 75 down to the immediate left of the 6 we wrote down earlier. At this point, the answer should look like this: 756.

So, 12 × 63 should be 756, and Wolfram|Alpha verifies 756 as the correct answer!

The process does seem new and long, but it's not hard when you become familiar with it. As you practice, the steps will merge together, and you'll flow more easily from step to step.

Let's run through the process again, this time working out the answer to 17 × 63.

We start by mentally figuring 17 × 3 is 51, and we write down the 1 (the ones digit of 51). After figuring out the first answer like this, writing the ones digit down will quickly become habit.

51 doubled becomes 102, and then we add 102 (our newly doubled number) to 5 (the tens digit) to get 107. Write down this 107 to the immediate left of 1 already down on paper, and we have 1,071, which is the correct answer!

When writing down the ones digit, such as the 1 from 51 in the previous example, I like to imagine that the tens digit is also there, as if it's written in a special ink which only I can see. That way, when I double that number, I can look at the paper later and add this “invisible” number.

Eventually, the whole process should flow in a continuous, easy manner. Using 23 in the following example, I'll write it out as you should actually think of it in performance. The text in regular characters is what you would think, and the text in italics are actions:

“23 tripled is 69...*write down 9*...69 doubled is 138...138 plus 6 is 144...*write down 144 to the left*...DONE!”

At this point, you'd have 1449 on the paper, which is 23 × 63!

From the earlier examples, were you able to follow this “stream-of-conciousness” version? If not, go back and make sure all the steps are clear!

As I mentioned earlier, you want to become comfortable performing this with numbers up to 33. I suggest practicing by having Wolfram|Alpha generate random numbers with which to practice. You can then verify your answers with a calculator, or by having Wolfram|Alpha work out the answer (just change x to whatever number you used).

NUMBERS ENDING IN .5: To truly master the skill I'm eventually going to teach, you should also be able to handle numbers ending in .5. This might seem much more challenging, but it's really not.

Let's start with just the multiplication by 3. Take the number 10 and 10.5, for example. The former is easy to multiply by 3, as it's 30. 10.5 multiplied by 3 works out to 31.5. All you're really doing when there's a .5 involved is adding 1.5. A simple way to think of this is just to initially ignore the .5, and multiply the rest of the number by 3, then think of the next number followed by “.5”.

Let's try multiplying 12.5 by 3 mentally. Start by just multiplying 12 × 3 to get 36. The next number is 37, and we throw the .5 on the end to get 37.5! Get the idea?

In step 2, it says that you should only write the ones digit of the number down. This needs to be amended to include the possible .5 at the end. In our previous example, step 1 gave us 37.5, so this modified step 2 means you should write down 7.5 on the paper.

What is 17.5 × 3? 17 tripled is 51, the next number is 52 and .5, so it's 52.5, and then you'd write down 2.5.

Let's try multiplying 23.5 by 3 in a stream-of-consciousness approach similar to the 23 example we used earlier:

“23 tripled is 69...70...point 5...*write down 0.5*

Step 3 is, of course, doubling the number. Fortunately, doubling any number which ends in .5 will always result in a whole number.

When doubling a number which ends in .5, just double the whole part, then mentally jump to the next whole number. What is 37.5 × 2? The whole part is 37, which doubled is 74. The next whole number is 75, so 75 is your answer!

How about 52.5? 52 doubled is 104, and the next whole number, and the answer, is 105! By now, doubling 70.5 should seem to be almost no problem at all. 70 doubled is 140, and the next whole number is 141.

Now, for the first time, let's try running through the full process starting with 28.5 as the given number. I'll run through it in the now-familiar stream-of-consciousness style:

“28.5...28 tripled is 84...85 point 5...*write down 5.5*...85 doubled is 170...171...171 plus 8 is 179...*write down 179 to the left*...DONE!”

What's written on the paper now is 1795.5, which is 28.5 × 63!

Once you've got the full process down, including how to handle numbers ending in .5, it's time to practice. Use this link to half Wolfram|Alpha generate random numbers from 0 to 33, which may or may not end in .5. You can check your answers with the same link used above, or a pocket calculator.

PART 1 WRAP-UP: Practice this skill between now and this coming Sunday, and then you'll be ready to handle the next mental math shortcut. The doubled number you calculate in step 3 will also be useful as this next shortcut's starting point, so you're already a little bit ahead of the game!

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1 Response to Mystery Skill 1: 63 Times

8:36 AM

Hmmm, I'm curious as to where this is leading, looks like I'll have to practice my 63 times tables