I am Soooooooooo Behind

The further I progress in algebra, the harder it is to blog about my progress. Not because I don't understand what I'm doing -- I do! The further I go the more complicated the notation gets, which means either lots of sloppy computer shortcuts (I hate the look of x^2) or tedious html coding. Type polynomial. Go into html editor. Add <sup> tags. Find out I didn't close half of them. Go back and close them. Rinse, lather, repeat. I don't think I can begin to handle a post on matrices without going into Photoshop, drawing one, then using it as a template for all the steps.

I can hear you now. "Laura, for reals. No one reads this blog. Why put yourself through all this?"

1. Blogging reinforces my understanding of the material. It makes me organize my thought process.
2. Consider this a public copy of my notebook, which is already beginning to fray at the ends. And fill up. Seriously, I'm almost out of paper, and I haven't finished the book!
3. Laugh at the crazy woman who fills her spare time with algebra. And then one day your kid will come home with that terrifying textbook filled with x, y, polynomials, and WORD PROBLEMS. 

And then you'll remember that nutty friend of yours on Facebook who posted about algebra and feel (only a little) bad about mocking her nerdiness while you're frantically looking for her blog. All the while your kid is sitting there with a page of unfactored polynomials.

So yes, I am going to get all these posts done and up. Eventually. I'm taking a breather before I start complex numbers. As I vaguely recall that's where Algebra II hit the fan for me. Or maybe it was functions. The point is I'm going to use this pause time to catch up the blog. There's also a large divergence starting between Idiot's Guide and Everything. I'm not yet sure if it's simply different sequencing or if they're going in completely different mathematical directions. I think it's a good place to rest. The books are due February 8, and if I don't finish there's no rule saying I can't check them out again. Well, unless there's a hold on them.

COMING SOON!!!!

See...MATRICES!

Adding, subtracting, multiplying,

and introducing

DETERMINANTS

See...POLYNOMIALS!

Adding, subtracting, multiplying, dividing,

with special guest star

FACTORING

See...RADICALS!

Adding, subtracting, multiplying, dividing,

and FRACTIONAL EXPONENTS!

Systems of Equations

In my word problem slogging I accidentally jumped ahead and worked on this without even knowing it.

A system of equations means they share the same x and y values. The equations themselves are not equal to each other. There are three ways to solve them.

Graphing: Slap the equations on a coordinate plane to find your answer.

Pros: Visual confirmation of (x,y) values.

Cons: Good luck eyeballing the difference between 4.33 and 4.4

Naturally algebra books rig it so any problems you're instructed to solve by graphing will have nice round non-decimaled numbers.


Substitution: This is my personal favorite. Solve one equation for a variable (usually whichever's easiest to isolate), plug it in to the other equation, get a numerical value for a variable, then use that value to solve for the remaining variable. I made that more confusing that is really is, didn't I? Let me demonstrate.

Solve system x - 4y = 11

                      3x + 7y = -5

Since x has no coefficient in the first equation, let's solve it for x.

x - 4y = 11

x - 4y + 4y = 11 + 4y

x = 11 + 4y

Now take x = 11 + 4y and use in the second equation. No, you can't use it in the first equation. You'll end up with 11 = 11. Even I know you can't do that. How did I know that? Never mind...

3(11 + 4y) + 7y = -5

33 + 12y + 7y = -5

33 + 19y = -5

19y = -38

y = -2

Whether you want to solve for x in the first or second equation is up to you. The first one looks easier to me.

x = 11 + 4y (Why go here instead of standard form? Do you really want to move the ys around again?)

x = 11 + 4(-2)

x = 11 + -8

x = 3

The coordinate pair for this system is (3,-2)

Elimination Method: I get it, but I don't use it unless instructed. Multiply one or both equations by a real number, set it up like an addition/subtraction problem, and eliminate a variable.

Solve system 2x - y = 13 

                     6x + 4y = 4

Greatest common factor time! You can do either 4(2x - y = 13) or -3(6x + 4y = 4). Why -3? If you go that route your goal should be to eliminate the 6x, for which you'll need it's opposite. For some odd reason I went with -3 even though I walk the path of least resistance.

 6x + 4y =    4

-6x + 3y = -39

 0x + 7y = -35

y = -5

Solve for x

2x - (-5) = 13

2x = 8

x = 4

The coordinate pair for this system is (4,-5)

Where Did My X and Y Go, and Should I Care?

Sometimes variables disappear when solving systems of equations. Check your math FIRST before classifying it under one of these two exceptions.

1. If the ending statement is FALSE (7 = 5), the system has no solutions and their lines are parallel on a graph. If you're not staring down a classroom clock, graphing the equations is a good way to confirm it. If you wind up with intersecting lines, you messed up somewhere. Please don't ask me to figure out where.

2. If the ending statement is TRUE (10 = 10), the system has multiple solutions and is inconsistent. This means their graphs overlap and the equations are part of a dependent system. They'll never break up with each other because the crazy they know seems safer than unknown crazy.

Systems of Inequalities

Both Everything and Idiot's Guide solve systems of inequalities by graphing them, shading overlapping regions, and picking test points within the answer region to confirm. Everything tells you how to plot it on a TI-81. I don't have a graphing calculator, nor do I ever plan on owning one. If I recall correctly, learning how to program those suckers is almost as frustrating as learning the math. Sorry, can't be of much help here.

So...I have an admission to make. My blogging hasn't slowed because I've stalled in the material. Quite the contrary. My blogging's stalled because I got my daughter Lego Star Wars: The Complete Saga for her birthday. And by got it for her birthday, I mean bought it for us to play together under the pretext of a birthday gift. And by play together, I mean she watches me play because her motor skills aren't good enough to handle the controls yet.

Okay, fine. I like Lego Star Wars. I've only cared about two video games in my life. This is one of them. (The other is the original 8-bit Tetris, but only if I have to blow on the cartridge.)

Monday and Tuesday I waded through matrices and Cramer's sadistic Rule. Today I got through adding, subtracting, multiplying, and dividing (both long and synthetic) polynomials. The matrix post will take some time. It requires a lot of drawing in Photoshop and formatting. All About Polynomials should be up tomorrow.

Inequalities

Yes, I did start the matrix today! But first I need to write down what I've learned about inequalities before I forget it all.


Why have inequalities at all? Here are some examples of real life inequality situations -- mathematical ones, anyway. Most of us don't set up word problems when we think through them. We just make do. I only worry how to solve what is presented to me.

Second grade (guessing) math sign review

x < = less than but not including
x > = greater than but not including
x < = less than or equal to
x > = greater than or equal to

The "but not including" part isn't usually included when one first learns about greater than and less than. It is crucial to note the difference when graphing inequalities.

Rule #1 for inequalities is reverse the greater/less than sign if you multiply or divide the entire inequality by a negative number. Or the "faster, more intense" version:

REVERSE THE GREATER/LESS THAN SIGN IF YOU MULTIPLY OR DIVIDE THE ENTIRE INEQUALITY BY A NEGATIVE NUMBER.


Basically inequalities are solved the same way as equalities but without the equals sign.

Solve -5x + 3 > -32 for x.

-5x + 3 > -32
-5x + 3 - 3 > -32 + -3
-5x > -35
-x > -7
-1(-x > -7) -- Apply Rule #1
x < 7

Meaning x is any number less than but not including 7. 6.99 will work. 7 won't.





Compound Inequalities

Using only one sign isn't enough? You want to x to be greater than AND less than in the same inequality? Compound equations are for you. Consider it minimum and maximum speeds on the interstate. 45 < x < 70 where x is ticket-free speed. In algebra books x has a coefficient which must be dealt with.

Solve inequality -4 < 3x + 2 < 20

1. Isolate x.

-4 + (-2) < 3x + 2 + (-2) < 20 + (-2)

-6 < 3x < 18

2. Eliminate the coefficient.

(-6)/3 < (3x)/3 < 18/3

-2 < x < 6

On this hypothetical highway you can drive as low as and including -2mph and up to but not including 6mph.

Solving Absolute Value Inequalities

Absolute value is the Jessica Fletcher of algebra. Everything is nice, neat, and tidy in your police report head. Then here comes this well-intentioned mystery writer asking too many questions two little bars which upend all understanding of the murder inequalities. We could also go Columbo here with "just one more thing," but I'm a Murder, She Wrote fan.

But what if the murderer used an absolute value inequality? We use them all the time in Cabot Cove.

(Side note: This still is from "One White Rose for Death," Season 3 Episode 4. For those of us born too late and too far from Broadway to see the original production of Sweeney Todd, this episode is a fantastic opportunity to watch Len Cariou and Angela Lansbury work together. And in a later episode his character plays a priest. No meat pies, though.)

Why are you staring at me like that?

Fine, fine, back to the nerdiness you signed up for.

If the absolute value is on the LESS THAN side:

Solve the inequality |2x - 1| + 3< 6

1. Isolate absolute value portion on the left side of inequality. If it's not already there, rewrite it that way (e.g. -- 6 > |2x - 1| + 3 = |2x - 1| + 3 < 6). At least that's what Idiot's Guide says. It may not matter that much.

|2x - 1| + 3 - 3 < 6 - 3

|2x - 1| < 3

2. Drop the absolute value bars and create a compound inequality with the opposite of the "greater than" integer. If you set things up correctly it's the number to the right of the < sign.

-3 < 2x - 1 < 3

3. Solve the compound inequality

-3 + 1 < 2x - 1 + 1 < 3 + 1

-2 < 2x < 4

-1 < x < 2

Even though this example uses <, < uses the same process.

If the absolute value is on the GREATER THAN side:

Here's where it turns out there are two murderers answers. It gets complicated. Stay with me until the trap is sprung.

Solve the inequality |2x + 5| - 4 > -1

1. Isolate the absolute value just as you did with "less than."

|2x + 5| - 4 > -1

|2x + 5| - 4 + 4 > -1 + 4

|2x + 5| > 3

2. Split this into TWO inequality statements. One simply drops the absolute value bars. The other one drops the absolute value bars, flips the sign to "less than," and negates the integer. Put an "or" between them.

2x + 5 > 3     OR     2x + 5 < -3

3. Solve each inequality separately.

2x + 5 - 5 > 3 - 5     OR     2x + 5 - 5 < -3 - 5

2x > -2     OR     2x < -8

x > -1     OR     x < -4

In summation:

|x + a| < b = ONE answer in the form of a compound inequality

|x + a| > b = TWO answers with an OR statement between them

Graphing inequalities

If graphing on a number line open dot means "but not including," filled dot means "and is equal to."

If graphing on a coordinate plane, dashed line means "but not including," solid line means "and is equal to."

Absolute value inequalities will have V-shaped graphs, not lines.

I have no desire to draw examples in Photoshop. Go look at someone else's work. Number line, coordinate plane

Tomorrow I will explain matrices as best I can.

And I'm Stuck

Word problems FTL.

I downloaded worksheets from Kuta Software for more practice material. Systems of equations went well. Mixture problems required some Googling. Once I got a clearly worded example I was set. Everyone has a tendency to do it with things they understand well, but super math minds go all George Lucas when asked to explain their work.

What do you mean you don't see your motivation? It's right there. I wrote it.



When I got to the rate of work problems my decoder skills hung on through "calculate how long two people can do a job together if A works at x rate and B works at y rate. Then came "If A and B work together at x rate, and B works alone at y rate, how long will it take A to work alone?" It should be a simple matter of taking the equation I was already using and solving for a different variable.

And yet it is not proving so simple.

Working together, Paul and Daniel can pick forty bushels of apples in 4.95 hours. Had he done it alone it would have taken Daniel 9 hours. Find how long it would take Paul to do it alone.

I set up my equation

(40 bushels / x hours) + (40 bushels / 9 hours) = (40 bushels / 4.95 hours)

My first thought it to cancel out all those 40s by multplying by 1/40, which gets me to

1/x + 1/9 = 1/4.95

Okay...I still have fractions. Which would be totally okay IF 1/4.95 didn't translate to .202020202.... Time for the homework helpers at algebra.com. (Tip: If you have a child enrolled in algebra who aces their homework but bombs tests, block this site and others like it. Actually just block it until you need to check their homework.)

One solution pulls the number 44.95 out of their Lucas machine, plugs it into the equation, and gets the answer. Where did that number come from?!?!?!?! Another solution explains it by having me multiply across the equation by 9(4.95). Which...I guess that makes sense because I want to get everything to a denominator of one. Why couldn't I see this yesterday? Ugh. The library computer is flashing that my session ends in ten minutes, so I'll skip the work and tell you the answer is 11.

Must post about inequalities before Monday. Must must must. Matrix starts Monday at all costs. Books are due February 8. Worst case scenario I check them out again once they hit the shelf. I doubt anyone wants to fight over algebra books.


Update: After my library session timed out I finally had that epiphany I was searching for. It wasn't multiply everything by 9(4.95). It was cross multiply everything by 9 * 4.95 * x, canceling out as appropriate. So yes, 44.55 came from somewhere, but I couldn't figure out how it went from one x to two.

For my own reference when Sue is having an algebra headdesk moment:

1/14/13 -- I just realized those equal signs in the middle should be multiplication signs. I spent so long trying to remember how to draw lines in Photoshop I forgot what kind of lines to draw! I'll fix them. Someday. Until then use your imagination.

equals 44.55

 equals 4.95x

equals 9x

SO

44.55 + 4.95x = 9x

44.55 = 4.05x

11 = x

Halle-frickin-llujah!

Somethings don't translate well from paperwork to computer notation. And for that there is Photoshop.

Crop tool only works on hands

Word Problems as They Should Be Written

It's that time -- WORD PROBLEMS

Dun, dun, DUN!!!!!

Word problems truly are Idiot's Guide's sole deficiency. The only word problems are in the last chapter and deals more with d = rt as opposed to "If Mary has three apples..." I don't need practice with d = rt! I remember that one! I don't remember divining how many apples Mary has out of no practical information! Do I look like Shawn Spencer to you? "Luckily" Everything is rife with them. I rewrote them with a fresh side of snark.

1. Kristen's grandfather was a college math professor for many years and thinks people enjoy his little challenges. He keeps a jar of coins with only quarters, nickels, and dimes. One day Grandpa tells eleven-year-old Kristen, "There's $48.50 in my silver coin jar. The number of Roosevelts is 20 more than the number of Washingtons, and there are 30 more Jeffersons than Roosevelts. If you can figure out how many of each coin there are by the time your mother gets home, you can have the jar."

Real life answer: Kristen waits until Grandpa takes a nap, takes the money, counts the number of coins, and pays the neighboring unemployed CalTech grad in WoW tokens to come up with the needed equation. Grandpa keeps his word and gives her the jar -- but not the money. Because Grandpa is a jerk like that.

2. Sonya's idiot assistant forgot to mail a very important contract to a client. She also didn't pay the internet bill so email is out of the question. Now Sonya has to take it to the client who is 6 hours away. She gets on the interstate at 7 a.m. and sets her cruise control at 65 mph. Minutes later Allison, aforementioned idiot assistant, realizes she forgot to put the contracts in Sonya's briefcase. She flies out the door with the contracts, jumps in her car, and sets off after Sonya. She gets on the interstate at 7:30 a.m. What is the lowest speed she can set her cruise to catch up with Sonya in less than two hours and hopefully keep her job?

Real life answer: Allison catches up with Sonya but is pulled over by a cop as they're exiting the interstate. Allison gets a $150 ticket. Once Sonya secures the contract in her briefcase she fires Allison. As Sonya drives away she neglects to realize she left the briefcase sitting on the road beside her car. Allison snatches it, throwing it in a drainage ditch on her way back home.

I'm running out of snark steam.


3. Xiang, Amy, and Jiang are 3 years apart in age. Xiao is the youngest and Jiang is the oldest. If 30 more than the sum of Xiao's age and Amy's age is 3 times Jiang's age, can Amy vote in the U.S. general election? 

Real life answer: Amy is old enough to vote in the US general election, but she won't because she thinks it's pointless. Great civic attitude, Amy.

4. Juanita opens a pottery studio in an upscale area of town. Her expenses for total 15n + 320 per day when she has n pieces of pottery to sell. If Juanita charges $28 per piece, a totally realistic price for handmade pottery, how many pieces must she sell each day to make a profit of at least $100 per day?

Real life answer: After the 1000th customer who (1) claims they could totally make that pot for a third of the price, (2) tries to haggle her down to Walmart prices, and (3) allows their children to break pieces and refuses to pay for the damage, Juanita stages a kiln accident that consumes her entire studio for the insurance money.



Actual Mathematical Answers

1.) Solve for quarters. BUT any expression of x must be multiplied by the value of the coin the expression represents. And despite the book examples don't use q, d, or n. I couldn't get past the coin value using variables representative of the coin names.

#quarters = x
#dimes = 2x + 20 (20 more than 2 times quarters)
#nickels = 2x + 50 (30 more nickels than dimes)

25cents(x) + 10cents(2x + 20) + 5cents (2x + 50) = 4850cents
25x + 20x + 200 + 10x +250 = 4850
55x = 4400
x = 80 quarters (or $20.00)

Now solve the other expressions for x

2(80) + 20 = 180 dimes ($18.00)
2(80) + 50 = 210 nickels ($10.50)



2.) I actually handled this one mostly by myself. Probably because I see it as a physics rather than an algebra problem, and while I don't take road trips I drive a lot. I have to figure out how far Sonya travels in two hours AND how far she travels in her 30 minute head start (expressed as .5 hours) to account for it in Allison's catch up speed.
d = (65)(2)
d = 130mi in 2 hours

d = 65(.5)
d = 32.5mi in .5 hours

The formula for Allison's catch up is what I had to double check. I wasn't completely sure where that head start factored in.

130 =  2r - 32.5
162.5 = 2r
81.25 = r

The book gives this as the accepted answer. I consider the actual answer 82 since (a) Allison needs to catch up in less than 2 hours and 81.25 will take exactly 2 hours (perhaps this is best expressed as 81.25 < r) and (b) how are you going to set your cruise somewhere between 81 and 82 miles?



3. Everything is nice enough to prompt me to solve for Amy's age (a), which makes sense because she's the middle of the group. I know Xiao is three years younger (a - 3) and Jiang is three years older (a + 3). Now what to do with that whole 3 times older than yada yada yada and how to set it up.

[Amy's age (a) + Xiao's age (a - 3)] + 30 = 3(a + 3) Jiang's age

a + (a - 3) + 30 = 3(a + 3)
2a - 3 + 30 = 3a + 9
2a + 27 = 3a + 9
-a = -18
a = 18



4. I totally blew this the first try.

15n + 320 ≤ 28n + 100
15n ≤ 28n - 220
-13n ≤ -220
13n ≤ 220
n ≤ 16.92

Even at $28 a pop that doesn't feel right. *look up answer* *go back in chapter for example* The $100 should've gone with the expense formula, not the retail price. And I don't need the ≥ sign, only the >? But it says at least which usually translates to "equals to"...fine. Whateves.

28n > 15n + 320 + 100
28n > 15n + 420
13n > 420
n > 32.30 or 33 pieces per day



I have to google more problems to work. Only the physics-ish one came naturally.

No Wonder I Was Making Great Progress

I've been saying for days I was about to start the matrix, right? Monday night after I posted my blog, ate some cake, and pondered leaving my five year old on someone's doorstep (it was a bad Mama day) I cracked open Everything to practice what I've done so far. My planned quick practice turned into three hours of BWUH? because Everything just had to be written by someone just like my Algebra I teacher. I didn't close the book until 3:30 a.m.

Idiot's Guide is amazing at breaking things down, but so far it hasn't thrown the same type of trick problems at me. The things my teacher would've said, "I just taught you how to do this with an example -- apply what I just said to something that looks absolutely nothing like what we just did!" Fractions with equations in both the numerator and the denominator. Equations with mixed numerals. The dreaded word problems.

I knew it was all going too smoothly. I'll give myself this much -- I must understand what I'm doing now. I was able to figure out which concepts applied to the equations.


(5x - 4) / 7 = (2x + 9) / 3

I saw this and thought, "What the !@#$ am I supposed to do with this shenanigans? I guess I could make the denominators match."

[(3)(5x - 4) / (3)(7)] = [(7)(2x + 9) / (7)(3)]

(15x - 12) / 21 = (14x + 63) / 21

Multiply by the reciprocal to get rid of the denominators... while writing this I realized I could've just cross-multiplied the whole thing from jump street. This is what I get for doing algebra at 1:30 a.m. I can't believe I did all that extra work. And I did it for three different problems!!!!

15x - 12 = 14x + 63
x = 75



When I got to (2n + 9) / (5n - 2) = 1/3 I freaked out a little. Bearings were quickly restored via the long method. Let's try it the short way with cross-multiplication!

3(2n + 9) = 1(5n - 2)
6n + 27 = 5n - 2
n = -29

No algebra after midnight for me. I added three more steps to this.


The inequalities were really disappointing in Everything. No "less than/greater than or equal to" dilemmas at all. I flipped through and didn't see a section on them at all. Everything Guide to Algebra my rear. There was one answer that threw me. I won't bother with the whole process because I did get it right. The book gave the answer as -6 > x > -18. Idiot's Guide calls this bad math grammar and would have me write it -18 < x < -6. I like that better because it's ordered as the numbers appear on a number line. Is one considered technically correct? Technically correct is always the best kind of correct.

The matrix keeps getting shoved back, so I'm officially setting a new start date of 01/14/13 so I can get my posts about fighting with word problems and inequalities up and not do so much of this back and forth between concepts. Books have to be renewed by the 17th, so I have to finish these books three weeks after that. This will get interesting.

A musing on how my most hated subject forged one of my most awesome friendships to date

Today on the *mumble*th anniversary of my birth, I break from my studies to reflect on one of the few good memories from math class.

I mentioned in my first post Algebra II brought me both a C and a good friend: +Betsy White , mathematical mind extraordinaire. She has umpteen places of pi wrapped around her office as a border, finds four leaf clovers with ease, and shares a love of froggy boots with my kid. She's taught calculus to high schoolers and currently teaches college engineering. I am green with envy at her ability to make sense of math problems with so many Greek letters I think they're actually directions to the Oracle of Delphi. Maybe they are and she's helping the math world keep it a secret from me.

Betsy and I already knew each other from marching band as fellow flute players, but we did not get off to a good start. Why? I don't remember. Even if I did, rehashing everything 15+ years later seems sort of tacky. Probably normal teenager/high school/"we're still learning who we are and screwing it up all over the place" type stuff. I can say that last one definitely applied me more than her. Actually that still holds pretty true for me.

*pause for introspective moment*

Betsy was a year behind me. Usually Algebra II is taken sophomore year. She had some sort of schedule snafu between first and second semester. Since she'd already taken (and I'm sure got an A in) Geometry, she wound up not only in my class but sitting right behind me in the Band Geek Corner of the classroom. Call it fate, divine intervention, a life lesson, but somehow it was Algebra II that made us close. Just as I don't remember the specifics of our initial feud, I also don't remember how that changed. That part should be revisited now. I hate that I forgot it. Metaphorically dragging me through the class by my hair definitely changed things. She burned the quadratic formula so deeply into my brain I remembered it until childbirth and parenthood wiped what little RAM I had left. I bet she carved it into the cheese of those Lance crackers she shared with me. All I know is we exited the class as friends despite the fact we were -- and still are -- pretty different people. Somewhere in that semester we found a way to understand one another.

We grew pretty close throughout my last two years of high school and stayed in touch into my junior year of college. And then we drifted. No catalyst, just life happening I guess. It didn't help I had recurring issues with depression and went down a hidey hole for several years. That's stuff for a different blog on a different day.

Three years and one day ago (I looked it up) we reconnected on Facebook. She still lives out of state but her parents are local to me. That summer we met for lunch. I won't speak for her, but I know on my end it was falling off a log. We were older, definitely smarter, debatably wiser, and while it took a bit to find our old rhythm it was still there. The then two year old that tagged along with me had a way of delaying things. Still does at five. Since then we've stayed in touch mostly through Facebook with a luncheon here and there when she's in town and our schedules mesh. Sadly they often don't. There's that real and complex life again.

I've been trying to wrap this post up for an hour with some deep insight wrapped in wit. It keeps eluding me so I'll end it sincerely. Thank you, Betsy, for both literally and figuratively prodding me through Algebra II. I got something besides a headache out of it after all.


Do you have anymore of those crackers?

Diving in Head First, or Equations I Did Not Miss Thee

Backtracking a bit to the beginning of my reintroduction to algebra. I can't get any work done on weekends.


With great enthusiasm and apprehension I opened Idiot's Guide to chapter four, the first appearance of basic equations. Surely I could start here, right? I flipped to the end for a test problem.

Solve the equation 9x + 3y = 5 for y


Perhaps the time honored adage "begin at the beginning" was in order. Back to pre-algebra.



Chapter one was filled with dusty concepts.

Mathematical classification systems: Both books went through the whole number hierarchy, but Everything had a Venn diagram nightmare implying all numbers were real numbers. Idiot's Guide defined real numbers as "all rational and irrational numbers" but didn't comment on the existence of "unreal" numbers. Thankfully there is now Google to tell me the counterpart to real numbers is complex numbers. It will be a hot minute before I feel like dealing with i.

Absolute value: Totally forgot about this. Whatever is inside the lines will be positive. Okay.

Associative and commutative properties: Order doesn't matter in addition and multiplication. One can cheat subtraction by changing 5 - 7 to (-7) + 5.

Identity and inverse properties: Adding 0 and multiplying by 1 do nothing. Adding the inverse negative gets 0, multiplying by the reciprocal gets 1. Forgot the fancy names but did remember the concept.



Chapter two was a skim through. All about fractions. Never had a problem with them. I did have to re-read the paragraph on division (multiply by the reciprocal). I wish I'd remembered that two weeks ago when I was struggling to divide 2/3 by 2 in my head. Yes, I felt dumb when .3333... popped up on the calculator. Very, very dumb.



Chapter three had the first appearance of the dreaded variable x. With exponents attached. Real and complex life awaited me, so I closed the book for the day.







The next day I took a deep breath and went back to chapter three. Idiot's Guide puts variables in a cross-disciplinary form I wish somebody had told me at the beginning.

Variables are pronouns.

Variables are pronouns!

Mind blown.

Does this make numbers nouns, operations verbs, and coefficients adjectives? Hold on, this metaphor is getting way more involved than I want it to. For now. Give me six weeks and I'll come up with a way to diagram a (very simple) math problem like a sentence. Or I'll Google around and find a meme where someone else did it. I can't be the first person to have this thought.

The rest of the chapter didn't have such strong epiphanies. Exponents were familiar but rusty.

• Add powers when multiplying
• Subtract powers when dividing
• Multiply exponents when the exponential expression is raised to another another power
• Negative exponents are considered bad grammar and must be written in reciprocal form.

Scientific notation: Positive exponents = Big number. Negative exponents = small number.

Distributive Property: This was where I got myself in trouble a lot in Algebra I...and II...and Precal. It's a wonder now I even made it so far in math. 5(x + 1) = 5x + 5 not 5x + 1. I now rewrite that expression as (5)(x) + (5)(1) in my work so I don't keep making that mistake.

Order of Operations: I sort of remembered it but sort of didn't. Parentheses first? Yup, still got it! Exponents? Uh, yeah sure. That totally came next. I did remember multiplication and division came before addition and subtraction, however by now I'd fallen into a very common trap that you do the multiplication and division in the order you encounter them, not all the multiplication first followed by all the division. I'm sure I knew this when I was actually in math class. Now not so much.

In Which I Connect Algebra to Schoenberg

Time to find equations based on perpendicular and parallel lines. It's easier than I thought but not as simple as I tried to make it.

The rules:

• Perpendicular lines have opposite reciprocal slopes. Idiot's Guide explains this as if line g is perpendicular to line h, m = a/b for g and m = -(b/a) for h. That wonked my head. I need an example I can process easier. Hello there, twelve tone row matrix! Opposite reciprocal = retrograde inversion. (Why, oh, why did I not have this connection to make the first time around?)

• Parallel lines have the same slope. Of course they do! They're parallel! Otherwise they'd intersect at some point. Maybe not on the part of the graph you're looking at but eventually.

My practice problem for perpendicular slopes is

Write the equation of line k in slope-intercept form if k passes through (2,-3) and is perpendicular to the line with the equation x - 5y = 7.

I can do this, right?

First solve for y.

x - x - 5y = -x + 7

-5y = -x + 7

(5y/5) = (-1x/-5) + (7/-5)

y = (1/5)x - (7/5) = slope-intercept of the line perpendicular to k

Now I need to take the retrograde inversion opposite reciprocal to get the slope of k.

(1/5)(-1) = -5 = m of k

On to point-slope form

y - (-3) = -5(x - 2)

y + 3 = -5x + 10

y + 3 - 3 = -5x + (10 - 3)

Which leaves me with slope-intercept

y = -5x + 7

Self-assigned extra credit -- write in standard form!

5x + y = 7

I am so proud of me. :)

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Then we come to parallel lines.

Write the equation of line j in standard form if j passes through (-6,1) and is parallel to the line with the equation -2x + 6y = 7

I can figure this out on my own. I don't need no stinkin' book!

...switch the signs? *check answer* I am un-proud of me.

Put in slope-intercept form to solve for y

-2x + 6y = 7

6y = 2x + 7

(6y/6) = (2/6) + (7/6)

y = (1/3)x  + (7/6)

Plug the point given in the problem into point-slope

y - 1 = (1/3)[x - (-6)]

y - 1 = (1/3)(x + 6)

y - 1 = (1/3)x + 2

y - 1 + 1 = (1/3)x + 2 + 1

y = (1/3)x + 3 = slope intercept of j

Convert to standard form

-(1/3)x + y = (1/3)x + -(1/3)x + 3

-(1/3)x + y = 3

[(3/1) * -(1/3)x] + [(3/1) * (y/1)] = 3(3)

-x + 3y = 9

(-1)(-x) + (-1)(3y) = (-1)(9)

x - 3y = -9

Note to self: always read the directions.

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I've already done the chapters on inequalities, but I will write about those over the weekend. Maybe I'll be able to finally sit down and write the 3 chapters I did pre-blog as well. Monday starts my descent into the matrix sans Neo. Wish me luck!

Graphs, graphs, graphs...

I was going to do a play by play of the chapters I've already done, but I'm afraid of forgetting my current epiphanies. I'll just back date those entries.


Today I covered chapters 5 and 6. Chapter 5 is the basics of graphing and graphing equalities. In a nut shell I learned graphs represent all possible solutions to an equation. I'm sure at one point I knew that, but today it was an "a-HA! My math teachers had a reason for this other than torturing me!" Basically instead of randomly choosing an x or y and plugging them in an equation -- my time honored method -- you can graph the equation and see which  x and y work together. I only see intercepts easily. Maybe that's all I really need to see well. Absolute value was a looooooooooong forgotten concept. Use it to find the vertex of a V-shaped graph. Got it. Only one of the example problems made me head tilt.

What coordinate pair represents the vertex of the graph of the equation y = -|x - 4| - 5 ?

First I solve the equation inside the absolute value for x. (I circle the original operation in my work so I don't confuse which one goes on both sides. Here it's highlighted in red.)

x - 4 = 0
x - 4 + 4 = 0 + 4
x = 4

So now

y = -|4 - 4| - 5
y = -|0| - 5

...what the !@#$ do I do with this negative sign outside the absolute value bars when my answer is 0? There's no negative zero like there's no crying in baseball! My pea brain decided since the 0 disappears I should be left with

y = - - 5
y = 5

The actual answer was y = -5. So the negative sign should've disappeared with the 0. Okay.

But why was that stupid negative sign there in the first place?


Chapter 6 was formula overload. I actually wrote them all in the first page of my fresh new notebook so I could keep track.

Formulas and concepts (re)learned

Slope formula

• (d-b) / (c-a) = m

• m = slope of line

• Positive slopes rise right to left.
• Negative slopes fall left to right.
• Horizontal slope is 0.
• Vertical slope is undefined because you can't divide by 0. Unless you get into that imaginary number stuff and I'm nowhere near the i chapter yet. That might even be a whole other book.

Point-slope formula

• y - y₁ = m(x - x₁)

• Good: Only one set of coordinates needed
• Bad: More opportunity for me to screw up multiplying all the way across the parentheses.

Slope-intercept formula

• y = mx + b

• b = y-intercept

• Good: x will always be 0!
• Bad: If there's a coefficient with y, there will be division. Must remember to divide on both sides of equation

I did all the practice problems. Most of my mistakes were forgetting to carry negative signs from one step to another, deciding 6/3 = 3, or just plain copying the equation wrong. Concepts solid, second grade math and reading comprehension sloppy. That's what my scientific calculator app is for.

Standard form of a line

This is the part where they tell me slope-intercept form is bad mathematical grammar. Correcting other's grammar in literary form with red ink is way more fun. *sigh* If the slope is a whole number it's golden. If the slope is a fraction I have my calculator ready. I solved this problem all by myself...until the last few steps.

Write the equation of the line that passes through the points (-3,7) and (4,1) in standard form.

This one uses all the goodies in the bag. I really do have to do every single step in the equation or I'll screw up on the shortcut. First calculate m.

m = (1 - 7) / [4 - (-3)]

m = (1 - 7) / (4 + 3)

m = -(6/7)

Now point-intercept. I chose (4,1) because they're both positive.

y - 1 = -(6/7)(x - 4)

y - 1 = -(6/7)x + [-(6/7) * (-4/1)]

y - 1 = -(6/7)x + (24/7)

(7)y -1(7) = [(7/1) * -(6/7)] + [(7/1) * (24/7)]

I am still adept enough with fractions to know when things cancel out.

7y - 7 = -6x + 24

Remember above when I said I circled the original operation in my work? This is the problem that made me start doing it. My original solution was

7y - 7 + 7 = -6x + 24 - 7 etc. y = 17

Flip to the back of the book. Not 17? Wait, 24 + 7 is...argh.

7y - 7 + 7 = -6x + 24 + 7

7y = -6x + 31

6x + 7y = 31

I got there. Wasn't pretty, but I got there.

My brain is a little fried from the endless formatting. My struggle with gleaning equations from lines perpendicular and parallel to the line you're looking for will have to wait until Thursday. My kid turns x - 5 = 0 tomorrow!

Did I just do that? I must be making progress!