Physically Stretching Angles To Get A Feel For Degrees

Units of measurement can be a tough concept, primarily because they are man made ideas.  Consider the time measurement of a "Minute." What the hell is that?  60 Seconds.  Well, what is a second then? Show me a second.  How about the length measurement of a "Foot." Common lore is it is a distance measured as the length of the Kings foot.  Damn, one could only imagine how weird that is as new kings come and go.  Anyway, these things are not endemic to nature, they are convenience grouping mechanisms (that are often not convenient!!)

Degrees are another such man made measurment construct.  We say things like circles have 360 degrees, or the corner of a square has a measure of 90 degrees.  Alas, what is a degree and how does one get a feel for what it is?

Making 90 degree corners is easy!

The girls and I were stretching for our daily Martial Arts session when we stumbled into a nice way to express our degrees of an angle.

The floors inside our house are big square tiles.  By having the girls do their stretching with  their butts on the grout line crossing points, we could line our selves up along the edges of the square to put ourselves at 90 degrees!

As we sat stretching at 90 degrees, I asked my youngest what half of 90 would be and she said 45.  I then asked my eldest to close her legs to half of what they were, and asked her what angle she was now making. "45 degrees dad"  Perfect!!

Show Offs! 180 degrees

Back to 90 degrees.  "How much is 2 x 90 degrees?" "180 degrees dad." "YES! Let's go!" and into full split action we went!!

Making Angles and Degrees an Experience rather than a theoretical mental construct aligns perfectly with our Homeschool Charter (III.Philosophy.4). Having them be the legs of an angle was key. They got a feel for various angle measures, how to measure the angle between two lines, and how four 90 degree corners could build a square; plus they did their stretching to boot!

As we wrapped up our stretching, the youngest piped up, "You know dad, it's like the arms on a clock"  Yup, sure is.

Discovering Math Tricks, Secrets, and Facts To Really Learn Algebra Mathematical Principles

"Dy, quick, what is 5 times 484,682?"
[about 1 second later]
"2,4,2,3,4,1,0 ....  that's an easy one dad ... give me toughie"

I love these kinds of interactions with my 9 year old daughter.   It allows her to show off her abilities, add to her confidence, flex her mental muscle, and work on her mental math skills.  But how did she know the answer so quick?  By applying a simple math fact*.

Cool as this is, what is more powerful and important is that an awesome mathematical fact (like the one she applied here) provides a vector for learning bigger, more important Mathematical principles.

To explain this more plainly, let's look at the math fact that Dy used.  You may have already figured it out, based on the way I wrote her response.

Here is the math fact she used:
5 times any rational whole number is simply that number divided by 2, times 10 (which is simply tacking on a zero if the starting number is even)

So, for her I made the problem easy.  First, I led off with "5 times".  At that point, Dy triggered into her 5 times facts.  Then I gave her a number that she could divide by 2 in her mind easily, simply going down the line of the even digits that made up the number 484,682.  So, she retained that number in her mind (a good mental exercise in its own right) and spoke out the result of dividing each digit by 2.  So, 4 ÷ 2 = 2, 8 ÷ 2 = 4, 4 ÷ 2 = 2, 6 ÷ 2 = 3, 8 ÷ 2 = 4, 2 ÷ 2 = 1, and times 10 so 0. So, 2423410.

She knows this fact now, which is cool.... and maybe during some test for college, a problem will come up with a 5 times in it and her speed will be helped if she recalls this fact .... but the real cool part is that in "discovering" this fact and how it works, she has nicely teed up algebra.

Here is the flow of her discoveries.

First, I challenged her to demonstrate her ability to multiply anything by 10 as I gave her a series of numbers (2, 4, 42, 544, 18, 20, 8, 532934, ....), which she did like a champ. Then I asked her to describe for me all the ways she could think of to write the number 10. I prompted her with "Like, 20 ÷ 2, or 10 x 1"  She paused, thought about it, and blurted "100 ÷ 10".  Awesomesauce! I kept poking her to produce more, and then she shared "2 x 5." Ahaha!  The one I wanted.

Second, I asked her to write down "4 x 10", which she did.  Then I asked her to replace the 10 with her example of the 10 equivalent, 2 x 5.  This resulted in "4 x 2 x 5."  We then explored multiplying in different orders, and to her amazement it always worked.  4 x 2 is 8 and 8 x 5 is 40, etc.

Third, I asked her if she could re-write 10 but with addition.  She wrote down 3 + 7.  I asked her to replace her 10, in her 4 x 10, with her 3 + 7,  which resulted in her writing 4 x 3 + 7.  I then asked her to solve, which yielded 19! (4x3=12, 12 + 7 = 19).  "Dad, that doesn't work." "No baby, it doesn't. The order you do this, the operation, matters."  We then launched into a nice discussion of order of operations, parenthesis, etc. for 30 minutes. :)

Fourth, I then asked her to go back to her original equation, 4 x 10 and write it out.
"4 x 10 = 40".  Next, I had her replace the 10 with her 2 x 5 version.  She wrote, "4 x 2 x 5 = 40".  Good.  Now I asked her "well, let's do the 4 x 5 first of the 4 x 2 x 5, since the order doesn't matter."  "That would be 20 then dad, so we have 20 x 2"  Me: "Well, that's interesting because 20 is half of 40, so that that makes total sense."

"Using this stuff, Dy, here is something that pops out that I'd like you to think about.   For any number that you multiply by 5, you could simply multiply it by 10 and cut it in half .... or cut it in half first and then multiply it by 10."  It wasn't instant that she understood it, but after 3 examples, she knew the trick.  After 10 examples, she was convinced. Trick learned.

Fifth, "Dy, let's replace the 40 with 20 x 2"  4 x 2 x 5 = 2 x 20  "Dy, do you see that both sides has the 2?  Does the equation work if we simply pull the 2 out?" It sure does.  

Sixth, "Let's re-write our equation but instead of writing the 2, let's replace it with a star"  Dy then writes out 4 x ☆ x 5 = ☆ x 20.  "I wonder if we can replace that star with anything?" Dy then starts trying numbers out.  2 works of course.  3 works.  How about 10.  Wow, she finds that everything works.  Neato!

Seventh, "We could always simply pull out that ☆ and it would be true right?" Dy then erases the star and sees 4 x x 5 = 20.  Big smile on her face.

A couple of wonderful hours with my little girl, and she now has an amazing foundation start with Algebra.

[Sidebar *: Early in this post, I used the phrase math fact.  I am using the word fact here because my wife hates when I call it a trick.  She likes to point out that it isn't a trick, just an application of some facts.  She is, of course, right (aren't they always! ;)  but many kids like to have secrets or tricks, and calling it a math secret could make your kid even more interested)]

You Suck At Math Because Your Parents Sucked At Math (ispo facto, your kid will suck because you do)

If your kid is awesome at mathematics, please don't read this post.  

If you're not interested in my rants, please don't read this post.

If you're a cry baby and take things too personal, please don't read this post.

If you're a believer in the victim mentality, please don't read this post.

Warnings done... proceed at your own risk.

Let's stop blaming the schools. Let's stop blaming the teachers. Let's stop blaming the genetics. Your kids suck at math because you do. This is true whether your kids are in public school, private school, or homeschool.

You suck at math because your parents sucked at math. Your kids suck at math because you suck at math.

"I can do math in my head, I know math" No you don't. You know how to drive a car, but that doesn't mean you know how it actually works ... and you need to know how it actually works to teach how to build a car.... and make no mistake, the fundamentals of math are all building block items. Multiplications, Fractions, Subtraction, Probability, etc.

"You don't need to teach them math, they will figure out what they need as they go." WHAT!?! Wow. How far does this line of reasoning go with you and into what areas of life? Or is this a selective thing only applying to math? Well, your kids, your choice. I, on the other hand, will most definitely ensure my children understand mathematics before they find they need it (such as with predatory credit card terms, mortgages, insurance rates, etc.)

So, who will break the cycle? You? Your Kids? Or will you continue to allow it plague your family lineage and build in an excuse system?

I was horrible in math in elementary school. The pain! The Pain! (I can hear Paul Mordeeb in Dune screaming in the pain amplifier!)
Memorize this, Memorize that, go faster, wtf is this? Crazy madness. It all made no sense because those who taught it didn't understand it. But wait, that's the teacher, right? Yeah, but parents are the first teachers. Teaching math is a natural part of raising a kid. Basic math is infused in all aspects of our society. I bet you don't understand it so you can't infuse it into a family discussion. I truly understand math. In fact, I can teach mathematics through set theory and numerical analysis (300 and 400 level college), and I know I can step into a college and pass a class in combinatronics.

Am I saying that my kids will be doing this level of mathematics before they leave home high school? Extremely doubtful. Unless one of my girls is very clear that she wants to go deep into some computationally intensive science, our math education will barely, if at all, touch trigonometry or calculus.

What is key is to really understand the fundamentals of computational mathematics and how it works. If your kid knows 6 times 7 of the top of their heads, that's fantastic. But does your kid really know what 6 x 7 means? Can they explain it, can they show it, can they swap in a something other than a 6 and still lay out what is going on? Can they substitute an apple for the six and still get it? Can they add the apple as the unit and get it? Can the explain what the units are implied or not by 6 x 7? Do they understand the place holders that the 6 and 7 represent? Do they really understand that we are simply speaking out, in a short phrase, "six times of a 7"? Do they know that 6 times 7 will never be presented to them that directly in the world? Can they transfer from a real life situation into the 6 x 7 memorized fact? Do they know the answer to 6 apples times 7 oranges? Do you? It isn't a trick question, if you understand mathematics.

If you don't really understand math, go back and study it. I did. As silly as this may sound to you, when I was 20 years old I decided I had to really learn math. So, while I was stationed in Korea, I plopped my ass down in the post library almost every day for six months and started to learn. I started with basic counting. Literally doing 1 plus 1. What was 1 plus 1 really? How does it work? What do these things mean? 1 what? Etc. Etc. I did this for about 6 months, getting into geometry. I now really know math and can transfer that into my children, making it a part of our daily lives. It isn't some vague concept or algorithm sequence to memorize. I can explain and show why, and I can infuse logical thinking steps into our daily lives to help my kids grow and explore through mathematics in real terms. I can have discussions around Miles Per Hour and help them discover how that is a fraction with units and translate that further into distance covered over some period of time. I sure wish I had this comprehension before entering the adult world.... I would undoubtedly have made better financial choices as a young man.

As a nation, we approach mathematics education wrong, and we seem to be suck in this vortex of wrongness. We can either let it perpetuate or fix it. I'm fixing it, what about you?

So, fellow educator, take accountability. Own it. Go back, and really think through mathematics at the basic level. Do it to gift your family tree, as I have.

Present Choices Activity

Here we are, a few days before Christmas. Presents under the tree, all looking seductive.  This gave rise to a fasctinating discussion around choices, mathematics, pragmatism and more.  Like so many of the good activities, this one evolved organically.

"Dad, can we open a present early?" - Dy (my youngest who is 8)

"Well, let's talk about this. I might be open to it.  Imagine if I gave you two options.
Option 1: You can open one present
Option 2: You can flip a coin, and if it is tails... you get zero presents, but if it is heads, you get two presents
Which option would you pick?" - Dad

--- Nice Long Pause --- The kind that shows the wheels turning in the mind

"Dad, I'd take option 1"  - Dy

A great 30 minute discussion then ensued by me changing the parameters.  For example:

"Dy, well how many presents would it take before you'd take option 2?"

and

"Dy, what if we used a dice instead and if you got an even number, no gifts... but if you get odd, you get two gifts?"

and

"Dy, what if we used a dice, and if you got a 1 or 2, no gifts, but if you get a 3 or higher, you get that many gifts"

and

"Dy, would it make a difference if we did this kind of deal 1 day before we normally open presents, or 5 days before we normally open presents?"

and

"Dy, would it matter if I get to pick the presents you choose from or if you do?"

and so on and so on.

As she would make a selection, we would talk about her reasons, the payoffs, the risks and more.  We then flipped it, and I asked her to offer me some version of the deal.  It was a very real feeling exercise because there were in fact real presents under the tree and she knew that whatever we settled on would indeed be offered up.

In our family, we've decided that the day before we normally open the gifts our girls will indeed be offered a deal.  It will be as follows:
Option 1: You get to pick one present from any of the presents with your name on it
Option 2: You flip a coin, and if it is tails you get nothing, but if you get heads, you get three of them

ho Ho HO!