Pointy Objects Like Triangles

I recently got to thinking about shapes. The curvy comfort of circles, and the calm, assured symmetry of squares and rectangles. The least comforting shape, though, was the triangle. Any way you hold it, you're not wholly comfortable. It pokes and shoves. Too many angles, students complain. Take the square- 90 degrees all the way. What's it with all these weird three sided things? They go this way and that- flat out and squat- and one obviously gets confused.

Save for all the complaints, the triangle is incredibly useful. In what?, you ask.
How about nachos? And the musical instrument, also called the triangle?
They are increasingly used in architecture, especially in bridges, and trigonometry is all about triangles, so, so incredibly important that even their peculiar shape shouldn't distract us.
So we'll talk more about this in today's post.

I assume you know how to differentiate between scalene, isosceles and equilateral types. If you have trouble with that, don't hesitate to ask, but for right now, we shall be moving on.
Let's talk about congruence. What is it, pray?
Congruence is nothing but the concept of things being the same shape and the same size. If there were two , lets go back to our famed doughnuts, chocolate doughnuts exactly the same size, and shape, they'd be called congruent in mathematical terms. Of course, we'd just call them delicious in real life though!
If there were two of you, like you and a clone of yours, wearing the same clothes, shoes, everything, you'd be called congruent, both of you.
Two objects are congruent if they are EXACT copies of one another. EXACT in bold. If your clone were wearing pink and you were wearing green, you wouldn't be congruent.

What we're primarily interested in now is the congruence of triangles.

Delving deeper into the number line

Hello again.
I can hear your groans, but please, I'll try to make it as interesting as I can.
Here's a picture of a..... doughnut?

Feeling better?
How about another one?

I'm actually hungry now. I'm going to take one away.

What happened here? Save for your scowls and hurt, what I did, can be expressed in mathematical terms:
I added a negative doughnut to the two doughnuts.

I added the additive inverse of 1 to 2.
2+(-1)
Think of the negative doughnut as a creamy, strawberry doughnut, different from the chocolate oozing one, but just as yummy (yummier? i think so!)

Okay then.
Next, what if I were to take away a 'negative' doughnut from the two former chocolate doughnuts, magically replaced?

2-(-1)

I subtract the additive inverse from the two doughnuts.
Imagine extracting a pink doughnut from thin air, imagine dragging it across the number line, watching it magically turn warm and chocolatey.
Voila, you have three doughnuts with you? Happy now? You should be.

Do not get confused. I'm trying to 'foodie' up the number line is all!

Envisage this:
-3-(-4)

Let's talk more about the doughnut analogy.
Take -3 in the number line. You need to subtract -4 from it, or add 4 to it. Both are THE SAME.

You have 3 STRAWBERRY doughnuts with you. (-3 is what you start with)
You must take away 4 STRAWBERRY doughnuts.( you need to subtract -4)

Take one away, you have 2 strawberry doughnuts.
Take another away, you have 1.
Take another away, you have, *sad face* 0.

What happens after this? We have one more step left.
Now is the interesting bit.
Think of the balance doughnut being dragged right of 0 in the number line.
You have one chocolate doughnut.

Confused? Try re-reading it for a bit.
Try this:

-7-(-12)?

You end up with +5, or 5 chocolate doughnuts. Think about it, and try to solve it on your own.

Remember, in your coursework, please refer to our chocolate doughnuts as positives, and the strawberry ones as negatives. The teacher wouldn't understand otherwise!

Somewhere over the.. number line

A few days ago, my friend came to me. Her sister, in the seventh grade, was confused with the number line concept, and would I explain it to her? I remember my days back when we were learning it. My teacher scribbling away at the board, and me, trying to catch up helplessly.

I did, eventually, but with difficulty.
So I thought I'd post about it, so it could help more confused souls out there.

The number line can be confusing. It's like an infinite lake, spreading in both directions like a confusing, endless thing. One misstep, and you think you've drowned. That is an exaggeration, but anything to liven this up, right? :-) I'm bad at jokes. 

Moving forward-

The start is easy enough. You have in one hand- a bunch of negative numbers, in the other a bunch of positive numbers and somewhere in between, maybe on your nose or forehead, a zero.

Imagine you're standing. On your left hand, imagine a magical ribbon shooting out. The longer the ribbon, the more negative the number. -10 would be a short one, -100 longer, -10,000 still longer, and negative infinity somewhere out there, endlessly moving forward.

Do the same on your right hand- out shoot the positives, smiley happy things shooting out.

Zero- bang in the middle.

Okay.

Please excuse me if you think I'm going too slowly. It's just that certain visualizations help a good deal. Remember- you'll be stuck with this seemingly innocent number line all your life- you wouldn't want to antagonize it. You'd be bickering for a l000ng time. See what i did there? ;-)

Now we know what the number line is, we've got an idea.

The next, irritating step is the movement. Add this, go right. Do this, slide left. Ah, the rights and lefts! The heres and theres!

Try to remember it like a dance- one of those fluid dances where the numbers are the glittery, fascinating dancers and the number line is the shiny, polished floor. Imagine you have a number, say 2. Add 3 to it. Where do you go from there? Left or right?

Think of 2 as a dancer. 3 is the number of steps she(or he, whatever pleases you), has to take.

Observe the sign. 3 is positive, so move the dancer '2' three steps to the right.

Is it confusing? I hope not. I merely referred to 2 the number as '2' the dancer.

Swish. Swish. Swish.

We reach 2+3=5.

What if the number isn't positive? Take -3.

Now the dancer '2' has to move three steps to the left. Another dance, perhaps a bit faster. They jump instead.

Jump. Jump. Jump.

2+(-3)= -1

here we are.

If you add a negative number to a negative number (in other words,  subtract a positive number from a negative one), what happens? can you visualize the dance?

-2 and -3:

-2+(-3)

The dancer '-2' moves three steps to the left.

-2-3=-5

Now, for an interesting topic-

The additive inverse.

Imagine you're standing in front of a mirror. You look into your image. It is you- it looks like you. But it is different. Imagine you're raising your left hand. Your image raises it's right hand.

What happened here? Lateral inversion in physics. You can think of it as an easy way to invert numbers.

The dancer '8' is looking at it's image '-8'.

The dancer '-3' is looking at it's image '3'.

Is it difficult to comprehend? I hope not. In doubt, think of yourself looking into the mirror. Think inversion.

Late, But Worth the Wait?

I have to apologize.
Since I started this blog two-and-then-some years ago (it has actually been that long), I've forgotten about it. I lack a more, what can I say, anodyne explanation, and that is the bitter truth.
*gloomy silence for a bit*
But now I've woken up from the seemingly endless hibernation, and we can start.

Now, since I'm new here and all (I have blogged a little, 'dabbled' as they say,but haven't blogged about maths or any particular subject),  I'll be starting, on a trial basis, by discussing a particular topic of my choice and solving problems related to it. When (if?)  more members join, I'll start addressing problems specifically. I'll try and put in a few video lectures too.

I think I'll have to get a-working and get a-stopping from rambling more.