## Posts tagged ‘calculator’

### Mental Math and the MCWC

How long would it take you to find the sum of 2 two-digit numbers?

What about 3 three-digit numbers?

Or 4 four-digit numbers?

Okay, let’s get really crazy… how long would it take you to find **the sum of 10 ten-digit numbers**?

You can decide whether you’ll do the calculation in your head, on a calculator, or with paper and pencil. Your choice.

With a calculator, it took me **91 seconds** to find the sum of 10 ten-digit numbers.

Without a calculator, the winner of the Mental Calculation World Cup 2014 needed only 242 seconds to complete 10 problems in which participants were asked to add 10 ten-digit numbers. On average, that’s just **24 seconds** to do in his head what took me a minute-and-a-half with technology.

**Holy smokes!**

Competitors at the MCWC do a number of mental calculation tasks. The following exercises will give you an idea of the computations that they do.

**Exercise 1.** Sum of 10 ten-digit numbers.

**Exercise 2.** Multiplication of 2 eight-digit numbers.

**Exercise 3.** Square root of a six-digit number.

**Exercise 4.** Day of the week for a calendar date.

**Exercise 5.** Multiplication of 3 three-digit numbers.

These sample exercises are taken from the examples in the MCWC 2014 Official Rules.

If you were able to complete all five of those exercises in, say, less than 3 minutes, then you might be ready for MCWC 2016. (Note that for Exercise 3, you needed to be accurate to within 5 × 10^{-6}.)

But if you’re like me, you’ll probably want to skip the competition and keep your calculator close at hand.

### Turn the Page

After eight fantastic years as the Online Projects Manager at NCTM, it’s time for my next chapter. On Monday, I become the Director of Mathematics for Discovery Education, leading a team that will build digital math techbooks for K‑12. I’m looking forward to building something great. As I mentioned during my interview, “I’m not coming to Discovery to create a textbook; I’m coming to create a *movement*.”

Leaving is such sweet sorrow. I’ll miss my friends and colleagues at NCTM, and I’m sad that I’ll no longer be creating resources for Illuminations. On the upside, my departure brought three stories worth sharing.

**A Day Off**

My last day at NCTM was February 28. That evening, I mentioned to my sons that I would not be going to work the next day. “Do you know why not?” I asked them. Alex suggested, “Because it’s Dr. Seuss’s birthday?” I love that! Celebrating the birth of Theodore Seuss Geisel certainly seems like a great reason for a federal holiday, but the truth is that I was just taking some time off between jobs.

**Lesson Learned**

The east coast was hit with a snowstorm during my time off, and both the NCTM and Discovery offices were closed. Had I been employed by either organization, I would have spent a day at home with pay. Instead, I spent an upaid eight hours designing the Vennebush Family Flag and playing Uno, Swish, and Qwirkle with the boys, while my gainfully employed wife dialed in to back-to-back-to-back conference calls. Moral: Check the forecast before quitting a job prematurely.

**A Parting Gift**

One of my colleagues at NCTM gave me a broken calculator. (And, no, this isn’t just a cheesy, elaborate set-up for a silly math problem.) The calculator used to be a normal, fully functioning, scientific calculator, but now it can’t add, subtract, multiply or divide without making an error. The good news is that the error is very predictable. The following video shows the results when using the calculator for four basic arithmetic problems.

The following (incorrect) results are shown in the video:

- 310 + 677 = 982
- 13 × 15 = 190
- 512 ÷ 64 = 3
- 75 – 10 = 60

And after the last problem, continual presses of the equal key should repeatedly subtract 10, but instead it shows consecutive results of 45, 30, 15, and 0.

Can you discern the pattern?

### Calculated Risk

If someone tells you that you don’t know what 1269 × 6 is, just tell them to go to…

Can’t hear the humor in that joke? Turn it up!

A few more calculator jokes…

What did the boxer do to the calculator?

He punched some buttons!

I have a digital calculator. Every time I want to count, I use my fingers.

Did you hear about the new calculator for dumb people?

It’s a giant hand with ten-thousand fingers.

“I can’t use this calculator,” said the student to his teacher. “It never does what I want it to. It only does what I tell it to.”

### Out of Sequence

If you came here looking for a good math joke and were thinking, “My God, I sure hope he didn’t post another story about his sons,” well, now would be a good time to close your browser.

Still with me? Good. Because I’ve just got to tell you about two things that have happened recently with Alex and Eli.

As we were walking through the neighborhood today, I was pointing to things that I thought the boys would enjoy. “Look,” I said as I pointed to the sky. Alex looked up. “An airplane!” he shouted.

I pointed to an automobile on the right. “A red car!” Eli exclaimed.

Then I pointed to a house on our left. Alex announced, “A square Fibonacci number!”

I admit, I didn’t see that coming. I had pointed to a house on our street, and there was a cat on the front porch that I wanted the boys to see. Instead, Alex saw the house number — 144 — and identified it as part of two different sequences. (I don’t think he recognized the lovely coincidence that 144 is the 12th term in both sequences, though I can’t be sure.)

My boys’ love of sequences is a result of teeth brushing. (No, really.) To make sure they brush their teeth long enough, I count while they brush — 15 seconds on the left side, 15 seconds on the right side, 15 seconds for the front teeth, and finally 15 seconds for “all around,” during which they’re supposed to brush the inside parts of their teeth. As you might well imagine, though, I was getting bored counting 1, 2, 3, …, 15 four times every night. I started to mix it up.

- I counted 1‑15 for the left side, then 16‑30 for the right, 31‑45 for the front, and 46‑60 for all around. But that got boring rather quickly, too.
- I switched to counting respectively by 1, 2, 3, and 4; that is, I’d count 1‑15 for the left, 17‑45 (by 2’s) for the right, 48‑90 (by 3’s) for the front, and 94‑150 (by 4’s) for all around. That was unsettling, though, because the boys started to think that 94, 98, 102, …, 150 were multiples of 4. While they recognized that counting by 2’s from 17‑45 gave the odd numbers, they weren’t able to discern that 94‑150 by 4’s analogously gave numbers
*n*≡ 2 mod 4. Nor did I expect them to — they’re only 3½. - I therefore switched it up again and counted by 1’s, 2’s, 3’s, and 6’s, which meant that “all around” was now 96, 102, 108, …, 150, which provided the more satisfying pattern of multiples of 6.

Then one night, Eli shocked me. He said, “Daddy, I figured out another way to count by 5’s.” I wasn’t really sure what he meant, but I assumed that “counting by 5’s” was a general term he was using to refer to skip counting. I asked him to explain. He said, “Like this: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.”

Upon hearing this, I thought what any geeky American father would think. “Holy sh*t. Did my three-year-old son really figure out the triangular numbers by himself?”

So I asked him, “What are those numbers, Eli?”

“Well,” he began, “there’s 1,” and then 1 + 2 = 3, and 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10, and…

Through a few more questions, I realized that he had ascertained these sums using my old TI‑83 calculator. I am proud to share a picture of the boys happily playing with technology:

The boys had been playing with some of my graphing calculators because they liked typing letters to spell words. Little did I know they were entering expressions and learning some math, too. (Take that, all you stodgy opponents of calculators in the classroom!)

After that, there were no holds barred; during teeth brushing, I started busting out all kinds of sequences:

- Triangular numbers: 1, 3, 6, 10, …, where
*T*(*n*) = ½(*n*)*(*n + 1). - Square numbers: 1, 4, 9, 16, …, where
*S*(*n*) =*n*^{2}. - Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …, where
*F*(*n*) = F(*n*– 1) +*F*(*n*– 2). - Feeby numbers: 11, 22, 33, 44, 55, …, where
*Fb*(*n*) = 11*n*.

(These are just the multiples of 11, but Eli named them the Feeby numbers, I think because it sounded a little like the first part of Fibonacci numbers, and two of the numbers in the list (55, 66) are elements of the Fibonacci sequence.) - Perrin sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …, where
*P*(*n*) =*P*(*n*– 2) +*P*(*n*– 3).

I’ve been thinking about sharing Moser’s Circle Problem with them and showing them the sequence 1, 2, 4, 8, 16, 31, 57, 99, …. Then again, maybe not.

Okay, so if you’re one of the folks who came for a math joke and tolerated that entire story, then by gosh you deserve a math joke. I don’t have a joke about triangular numbers (note to self: create a joke about triangular numbers), but I do have a joke that involves the word *triangular.*

What is small, green, and triangular?

A small, green triangle.

Yeah, I know… it’s lame.

Let me try to make amends with a cool triangular number problem.

Append the digit 1 to the end of every triangular number. For instance, from 3 you’d get 31, and from 666 you’d get 6,661. Now take a look at all of the divisors of the numbers you’ve created. What are the units digits of the divisors for every number created in this way? Can you prove that this result always holds?

I have a proof, but you’ll have more fun solving it on your own than reading my solution.