Multiplicative Dates

If you read the previous post containing Math Problems for 2024, then you already know what a multiplicative date is. But in case you missed it, here’s the definition again:

A multiplicative date is one in which the product of the month and date is equal to the (two-digit) year. For instance, today — January 24, 2024 — is a multiplicative date, because 1 × 24 = 24.

The question I asked in that previous post was, how many multiplicative dates will there be in 2024?

But I’m revisiting this topic again today, because today is the first multiplicative date of a year that has more multiplicative dates than any other year this century. That’s right, no year 20XX has more multiplicative dates than 2024.

Of course, there’s no reason to stop there. There are quite a few interesting questions related to multiplicative dates:

1. How many years this century have no multiplicative dates?
2. Which two consecutive years have the most multiplicative dates combined?

Finger Multiplication

A digital computer is one who adds on his fingers, and this definition reminds me of a quote by Tom Lehrer.

Base eight is just like base ten, really… if you’re missing two fingers!

Many students know the trick for multiplying by 9 on their fingers, which I mentioned in yesterday’s post. However, most folks don’t know the following finger trick for multiplying two numbers that are greater than 5 but less than or equal to 10.

1. On each hand, raise a number of fingers to indicate the difference between each factor and 5. For instance, 8 – 5 = 3 and 7 – 5 = 2, so to multiply 8 × 7, raise three fingers on the left hand and two fingers on the right hand, as shown below.

2. Count the total number of fingers that are raised on both hands, and append a 0. As shown above, there are 3 + 2 = 5 fingers raised, so this gives an intermediate result of 50.
3. Count the number of fingers that are folded on each hand, and find their product. Again using the example above, there are two fingers folded on the left and three fingers folded on the right, and 3 × 2 = 6. 
4. Add the results of Steps 2 and 3. For our example, 50 + 6 = 56, so 8 × 7 = 56.

We can use algebra to understand why this trick works.

Let L = the number of fingers raised on the left hand, and let R = the number of fingers raised on the right hand. Then the numbers we are multiplying are (5 + L) and (5 + R).

The total number of fingers raised is L + R. Consequently, the result from Step 2 is equal to 10(L + R).

The number of fingers not raised on the left hand is 5 – L, and the number of fingers not raised on the left hand is 5 – R. Their product, which is required in Step 3, is (5 – L)(5 – R) = 25 – 5(L + R) + LR.

Finally, combining them in Step 4 gives

10(L + R) + 25 – 5(L + R) + LR = 25 + 5(L + R) +LR = (5 + L)(5 + R),

which is the product we initially wanted to compute.

I recently learned one other method of multiplication. It does not require fingers, and I’m not even sure it’s useful, but it is interesting.

To find the product of two numbers, a and b, do the following:

Plot the points A(0, a), B(b, 0), and C(0, 1).

Draw segment CB, and then construct segment AD parallel to CB.

As a result, point D(d, 0) is a point on the horizontal axis such that a × b = d.

For instance, to compute 6 × 3, plot A(0, 6), B(3, 0), and C(0, 1). Then draw a line through A that is parallel to CB. This line will intersect the horizontal axis at the point D(18, 0), so 6 × 3 = 18.

A World Without 9

One of my favorite verses:

Once upon a time in our solar system,
We couldn’t make due without 9.
But Pluto’s not a planet now —
So 8 will due fine.

So sing the Barenaked Ladies in their wonderful song 7 8 9.

This got me to thinking: What would life be like if we didn’t have the number 9? It led to two questions that would be great fun to ask in a classroom:

• Imagine if 7 really had eaten 9, and the digit 9 were removed completely from the number system. That is, not only is the number 9 gone, but you couldn’t make 19, 92, or 1,090,394, either. What would our number system look like?
• Imagine if 7 had eaten only the number 9, but the digit 9 (or a suitable replacement) could still be used to make other numbers. That is, only the number between 8 and 10 is removed from the counting numbers. (Well, I guess technically we’d have to say that the entire interval [9, 10) is removed from the real number line, so that 8.999… = 10, but such a definition would go over the heads of most kids to whom I’d want to present this thought puzzle.) What would happen then? To think about it, calculate the results of the following problems:
1. 8 + 7 =
2. 8 + 14 =
3. 12 + 22 =
4. 4 × 5 =
5. 6 × 10 =
6. 11 × 12 =

As it turns out, my sons love this song. In fact, they love every song on Snacktime by BNL, and I have to admit that their daddy loves the album, too. Happily, my sons also love math, and they have shown a lot of interest in multiplication recently. A few months ago, Eli used the caculator to discover that 3 + 3 + 3 + 3 + 3 = 15, and he declared, “Daddy, if you add 3 five times, you get 15.”

“That’s right!” I said, and I explained that sometimes, instead of saying “add 3 five times,” we can just say “3 times 5.” Well, he thought this was extra cool. He entered 2 + 2 + 2 + 2 = 8 on the calculator and said, “Look, 2 times 4 is 8!” He has since converted thousands of addition problems into multiplication facts.

Just a few days ago, he made a statement that made my educator heart leap into my throat. He said, “Multiplication is just like adding a lot of times.” Of course, Keith Devlin might think this is a bad thing, but I thought it was a wonderful connection for Eli to make.

With their interest in multiplication sparked, we played with the Times Table on Illuminations. However, we did not use this as a modern-day set of flash cards with which my kids were supposed to memorize a bunch of unrelated facts. Instead, we talked about the patterns within the multiplication table, such as how the even numbers occur in the “times 2” row and that each number is found by adding 2 to the previous number. We also examined the square numbers, which occur along the diagonal.

Yesterday, I showed them the trick for multiplying by 9 on their fingers.

1. Make two fists. From the left, count as many fingers as the number you want to multiply by 9, and raise that finger.
2. Count the number of folded fingers to the left of the raised finger. This is the tens digit of the product.
3. Count the number of folded fingers to the right of the raised finger. This is the units digit of the product.

The example below shows the result when multiplying 9 × 7. The seventh finger is raised. There are six folded fingers to the left of the raised finger, and there are three folded fingers to the right of the raised finger. Consequently, 9 × 7 = 63.

(Lack of) Math Videos

I just saw a math video that showed a technique for multiplying two-digit numbers. I’ll share this trick with you in a minute. But I am dismayed that this video — along with so many other math videos I’ve seen on the web — are willing to show these tricks but not explain them. So I’m going to provide the trick, which I thought was pretty cool, along with an explanation as to why it works.

Before I do, though, I have to vent. Math videos on the web are a real problem. The ones I’ve seen generally do very little to promote conceptual understanding. Instead, the videos present skills without explanation, and the implicit message is that you don’t need to understand why it works — just do exactly as the videos say, and you’ll get the right answer. The issue is that students then see mathematics as a series of disconnected rules they need to memorize, instead of seeing it as the beautifully interconnected discipline that it is.

Arrgh.

Okay, I’ll now step down from my soapbox and share the trick with you…

Take two numbers, both between 10 and 99, that meet the following criteria:

• Their tens digits are the same.
• The sum of their units digits is 10.

Here are some problems that meet these criteria:

• 22 × 28
• 74 × 76
• 33 × 37
• 55 × 55

I will now state the obvious — there are only 81 multiplication problems to which this trick applies. Still, I think it’s fun to explore the math underlying the trick.

To find the product, then, follow this process:

1. Take the tens digit, and multiply it by one more than itself.
2. Multiply the units digits.
3. Finally, concatenate (put together) these two results to make a three- or four-digit number. (As noted in the comments below, if the result of Step 2 is a one-digit number, you’ll also need to put a 0 in front of it.)

Let’s take the first problem above, 22 × 28, to see how this works in practice.

1. The tens digit is 2, so multiply 2 by one more than itself: 2 × 3 = 6.
2. The units digits are 2 and 8, so multiply them: 2 × 8 = 16.
3. Consequently, 22 × 28 = 616.

More generally, the trick could be explained with symbols. If the tens digit of each number is n, and the units digits are p and q, then we’re trying to find the product of (10n + p)(10n + q). Using the FOIL method from algebra gives

(10n + p)(10n + q) = 100n² + 10n(p) + 10n(q) + pq = 100n² + 10n(p + q) + pq

But since the criteria implies that p + q = 10, this becomes

100n² + 10n(p + q) + pq = 100n² + 100n + pq

This can be further refined to

100(n² + n) + pq

and the important point, then, is realizing that n² + n = n(n + 1). Which is just another way of saying that we’re going to take the tens digit, n, and multiply it by one more than itself.

And that’s it. The first part of the expression above, 100(n² + n), gives the hundreds (and perhaps the thousands) digit, and the second part, pq, gives the tens and units digits.