Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the song, No. 1 wanted to show off, and said "Listen to me!"

Mom-person shut her down with "What's 3 times 2?"

A few seconds later No. 1 replied, "6!"

Getting three year-old No. 3 involved, Mom-person asked, "What's 2 plus 1?"

3 immediately, and happily replied "5!" her answer to almost all math questions.

Mom-person turned to No. 2 next, "OK, what's 2 plus 1?"

After a bit of work, and some reconnoitering of fingers, 3 responded with "3!"

I wanted to make sure 1 was keeping 1's hand in the game with binary numbers, and she was having fun demonstrating her math prowess, so I asked, "How do you write 3 in base 2?"

She thought for a bit, "11."

"Yup, that's it."

Then, Mom-person jumped in with the question that started up the rest of the conversation, "What's 10 times 10?"

No. 1 had in fact been working on memorizing her multiplication tables.  "100," she replied.

I asked her how you write 100.

"1, 0, 0"

1 and I had been worked on multiplication as shifting in binary a few weeks ago.  I hopped back in with "What's 2 times 2?"

"4"

"How do you write that in binary?"

"Yup!"

I hadn't known the pattern before, but I thought I saw it now.  I asked No. 1, "What's 5 times 5?"

No. 1 hadn't memorized this one yet, so she started in flashing one hand full of five fingers after another at herself, counting as she went.

Meanwhile, I was working through what base five digits would hold on my fingers.  You get up to four on the first digit.  On the second digit, 1 is equal to 1 in the fives place, you can all the way up to four though, so you can get four fives in the fives place followed by four ones in the ones place which gives you twenty-four before you run out of room which means.

"Five times five is 25!" No. 1 cut in just ahead of me.

Having just figured out everything would work out I asked, "How do you write that down in base 5?"

1 looked at me for a few seconds.

"Well, if 10 times 10 is written down as 1, 0, 0 in base 10, and 2 times 2 is written down as 1, 0, 0 in base two, what's 5 times 5 written down as in base 5?"

"1, 0, 0!"

And so we'd figured out an easy way to write down the square of a number, just write it down in its base, and it'll be, (you guessed it), 1, 0, 0!

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…