Monday, February 29, 2016

Doubling and Squaring, Cont'd

Can you show that the digits of 99...9 + 99...9 are the same as the digits of 99...9 × 99...9, discounting zero digits?

If you tried a few cases, then you observed a pattern:

9 + 9 = 18     and     9 × 9 = 81

99 + 99 = 198     and     99 × 99 = 9801

999 + 999 = 1998     and     999 × 999 = 998001

9999 + 9999 = 19998     and     9999 × 9999 = 99980001

Given a string of N consecutive 9's, if we double it, then we get a 1, followed by N − 1 consecutive 9's, followed by an 8. Whereas, if we square it, then we get N − 1 consecutive 9's, followed by an 8, followed by N − 1 consecutive zeros, followed by a 1. The digits of the double are the same as the digits of the square, discounting zeros.

Where does the pattern come from? Here is one way to look at the example of 9999:

9999 + 9999 = 10000 + (10000  2).

If we call the number in parentheses X, then we have

9999 + 9999 = 10000 + X.

This implies that the digits of the sum are just the digits of X preceded by a 1.


9999 × 9999 = (10000  1)(10000  1)

= 10000(10000  2) + 1 

= 10000X + 1.

Now we can see that the digits of the product are going to match the digits of the sum, discounting zeros, because multiplying X by 10000 preserves the digits of X (all it does is tack on zeros), and when we finally add the 1, the rightmost zero becomes a 1.

In short, the sum is of the form [1][X] and the product is of the form [X][0's][1].

I'll end with a general version of this line of reasoning.

Saturday, February 27, 2016

A Puzzle About Doubling and Squaring

Last night my younger asked: "Is 7 times 7 equal to 41? Because that's how it works for 9."

At first her question was mysterious to us. Then she explained that she knew that 9 + 9 = 18, and she had heard that 9 × 9 = 81, and she had noticed that in 18 and 81, the digits are reversed. So what she wanted to know was, does it work the same way for 7? Given that 7 + 7 = 14, can you conclude that 7 × 7 = 41?

It was a great question! But no, we answered, it doesn't work that way. A little reflection shows that 2 and 9 are the only positive integers for which doubling and squaring yield identical digits, up to ordering. (There will never be a multi-digit case, because squaring a multi-digit number always yields more digits than doubling it.)

However, 434 is an interesting case, because 434 + 434 = 868, and 434 × 434 = 188,356, so 434 has the property that the digits of the double are a subset of the digits of the square, counting multiplicity.

Another example like that is 99, because 99 + 99 = 198 and 99 × 99 = 9,801.

In fact, any number of the form 99...9 works this way. Can you show that the digits of 99...9 + 99...9 are the same as the digits of 99...9 × 99...9, discounting zero digits?

There are 4,747,374 integers k in the range 1 ≤ k ≤ 10^8 such that the digits of k + k are a subset of the digits of k × k, counting multiplicity. Of these, 83,238 are such that every digit appearing in k × k also appears in k + k.

P.S., Update just to add this graph, which is a scatterplot of (x, y) where x is a number between 1 and 10,000 and y is the number you get by reversing the digits of x. For example, the plot contains the point (46, 64) and the point (2539, 9352).

Monday, February 22, 2016

Oh, You

In each puzzle below, form a word by filling in the blanks with consonants.

 1.     _UO

 2.     OU_O

 3.     _UU_UU

 4.     _OU_ _ OO _

 5.     _ _OU_ _OU

 6.     _O_U_OU_

 7.     _O_ _UOU_

 8.     _U_O_U_O

 9.     U_ _ _UOU_

10.     _OU_ _ _OO_

11.     _U_ _ _UOU_

12.     _U_U_ _UOU_

13.     _O_U_ _UOU_

Saturday, February 20, 2016

In Defense of the New Met Logo

The Metropolitan Museum of Art (of which I'm a member) has traded their classical logo (above) for this one based on type:

The new logo has come in for a lot of criticism (NY Times, NY Magazine). I can see why, but I'm going to take my time over this one.

There's no question that the old logo was more authoritative—more "classical." It's also, I would point out, one hundred percent European, and more specifically Renaissance European—and that suddenly strikes me as odd, because if you've ever walked through the Met, then you know that this greatest of museums in the greatest city on Earth warehouses the artistic output of the entire planet from all times. So even given the centrality of Renaissance European art to the history of art, the museum's old logo just doesn't cover what the museum actually is.

Of course, one cannot claim either that the new logo reveals the museum's scope; the new logo is only type. But not attempting to show the scope is still more accurate than representing the whole by a part.

And I'm not so sure that the new logo doesn't allude subtly to the collection. Does the arch of the first T suggest to you a doorway in the Cloisters? Does the negative space between the E and T in the word MET suggest Picasso, or at any rate the artistic concept of negative space itself? How about the negative space in the H—an art deco vase? Does the M's shoulder wear the armor of a knight or a samurai? Perhaps type serves after all to hint at the chaotic variety of the Met. The compression of the letterforms and the exaggeration of their serifs combine to produce a vaguely helter-skelter feel, or perhaps the feel of a proto-alphabet like Phoenician.

I'm not yet convinced about the color, but I'll wait to decide until I see the print campaign. I think my strongest reaction right now is to the new logo's kinetic energy. What I mean by "kinetic energy" is that the word THE appears right-shifted in such a way that it seems to be trying to get ahead of things, or perhaps in such a way as to impart a sense of clockwise rotation to the logo as a whole. On a closer look, one can see that the letterforms in THE are in fact precisely aligned atop the corresponding letterforms in MET. But the fact that the logo begins and ends with T means—given the horizontal gap between the top of a letter T and its base—that the word THE still feels as if it slightly overhangs the word MET. Hence the kinetic energy.

The designer probably intends this effect: it suggests a museum in motion, whereas the previous logo suggests both solidity and stolidity.

Monday, February 8, 2016

Math Jokes!

For a long time, I thought "math joke" was an oxymoron.

Then I had kids.

At dinner, they say, "Dad! Give us a math joke!" Below are some that I've given them. Enjoy these denizens of the intersection (Dad jokes) ∩ (math jokes). I may update the list from time to time....

Q: When the Great White shark showed up, where was the octopus?

A: Octogone!

Q: If you cut up a general into 1,000 pieces, what is one of the pieces called?

A: A milli-leader!

Q: Where did the mathematician's family eat dinner every night?

A: At the multiplication table!

Q: Why were 10 and 11 mad after the race?

A: Because 20 won!

Q: What did 0 to say to 8? (I heard this one from Sophie, age 9.)

A: I like your belt!

Q: What did the pirate say to 5 × 16?

A: Ahoy, 80!

Q: What do da glasses help with?

A: Da vision!

Q: What do you call an angle after it gets in a car crash?

A: A wrecked-angle!

Q: What do you call a seagull who won't give up on her geometry homework?

A: A trying-gull!

Q: What do you have when you give a shot of anesthetic to a line?

A: A number line!

Q: What kind of snake likes to do math?

A: An adder!

Q:Why was the right angle sweating?

A: It was 90 degrees!

A mom said to her toddler, "I'm counting to three!  One, three!"  The toddler said, "Why did you skip two?" Mom said: "Because I can't even right now!"

A joke/riddle. Feel free to enter your solution in the comments!

ipip  ipip  ipip
ipip  ipip  ipip
ipip  ipip  ipip
ipip  ipip  ipip

P.S. Feel free to add your favorite math jokes to the comments. :-)

Tuesday, February 2, 2016

How We've Been Using Flashcards

Saturday School is never very long, but over time we still manage to do a range of things, including solving word problems, learning concepts, and doing exercises that build fluency and fact recall. Our materials include worksheets that I create, workbooks off the shelf such as Kyoiku Dojinsha and others, released test questions, homemade flashcards, and pennies or dice that we might use to play a math game.

Concerning flashcards in particular, a researcher in mathematics education saw this article of mine and later emailed me an important tip that I wanted to pass along:
[Flashcards] ... are a good fluency method [but] please stress that students should be spending most of their time on the cards they do not know yet or on those they know but are not fast on yet. Most drill uses many problems students know and thus is a big time waster.
A similar message appears in this document for educators (emphasis added):
Organizing practice so that it focuses most heavily on understood but not yet fluent products and unknown factors can speed learning. To achieve this by the end of Grade 3, students must begin working toward fluency for the easy numbers as early as possible.
Time is always scarce, and it is valuable to customize flashcard work to the student's current state of mastery. Some digital apps do allow you to adjust the settings so that students are focusing on the facts they need to focus on—maybe some of them even adjust themselves automatically over time. In any case, here's how I've tried to accomplish something similar using old-fashioned flashcards. I make no claim that my method is the best, or even better than others! But it has worked for us, and it's been fun.

Here's how it works:
  • The flashcards are kept in a "piggy-bank" made from an old tissue box.
  • We work at the dining table. My wife or I will draw a flashcard from the bank and show it.
  • If the answer comes back "lickety-split," then the card is set aside. Otherwise, the card goes back in the bank. If there is doubt, then the card goes back in the bank.
  • If the student is drawing a complete blank, then I prompt with a strategy, for example if the problem is 6×8 and the reaction is a blank stare, then I might prompt with "Do you know 5 × 8?" Then the student can say "Oh right, that's 40, so 8 more is 48." Of course, the card goes back in the bank, but it has been a good just-in-time learning opportunity.
We'll do anywhere from 5 to 15 minutes of this, and at the end, the student "owns" all of the cards that have been set aside. The student highlights the known facts on a map (images below) and puts the cards into a keepsake box.

This work proceeds in tandem with our other activities, including worksheet practice on facts (customized according to what the map says). Eventually, the day comes when there are no more cards left in the bank and the entire map has been highlighted. Time to celebrate! From then on we'll still do occasional maintenance practice to keep the facts secure (worksheets or a configurable app), and of course there are the worksheets that come home from school.

The way the system works is that the bank gets emptier and emptier over time, with known facts exiting the process as they become known. That's how the game exemplifies the advice I shared at the outset about customizing fact practice.

Although initially the bank contains many already-known facts like 2 + 1 or 3 + 0, this is intentional in order that the early sessions will feature easy, known facts and establish a foundation of confidence as we embark on the process. The rapid progress on Day 1 creates excitement. Pretty soon, those easy known facts exit the system and the bank becomes nicely focused on the student's individual horizon. Of course, this also means that the work is getting tougher over time, so I pay attention to motivation and emotions while we work.


Here are the two maps we've got going right now (addition and multiplication):

(The addition map is like the one I showed in this post.) We use magnets to pin the maps to metal shelves above the kids' desks. Each weekend, they chart new progress using a highlighter.

The maps are a great way to recognize accomplishment. The kids love adding to the maps and seeing their maps fill up over time. Completing the map brings a strong sense of satisfaction and achievement.

For my purposes, the map also suggests hypotheses about where prerequisite concepts might be lacking. I can address those directly in a separate line of work.

Here's what our flashcards look like. They are made out of ordinary 3x5 note cards. (To save paper, I write several problems on each note card and then cut them out with scissors.)

Now, the setup is actually a little different depending on whether we're talking about addition or multiplication. Let's consider addition first. By the time we're doing this, the student can mentally calculate all or almost all of the sums on the map, but some of the facts are very slow, and not too many of the facts are known from memory. So after I have written out all of the sums, I give the flashcards to the student and she goes through them one by one, writing each answer on the back of the card.

With the cards complete, we're ready for the funnest part: turning an empty tissue box into a bank. Here is a picture of last year's "piggy bank" of addition facts (piggy-posterior not shown).

Here are the containers they use to hold the facts they "own":

Multiplication differs from addition in the setup, because at the time when we first create the flashcards, the student doesn't really know how to calculate all of the products yet. Products only go into the bank if the student has seen them before and can calculate them mentally (perhaps slowly in some cases). I did a little probing at the outset to find out roughly where we were.

So that's how we've used flashcards in Saturday School. Readers might wonder why we only included sums and products, since students also have to be fluent with differences and quotients. Including differences and quotients is good, and I also like flashcards that show the entire fact family (you cover up two of the numbers, and the student tells you the third). In my case however, it happens that the kids are so secure with the relationships between operations that, presented with 14 − 9 or 24 ÷ 6, they just answer by consulting their mental "lookup table" of + or ×. They also get direct practice with differences and quotients in worksheets that I give them or that come home from school.

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