coordinate geometry

if twitter had been invented in the 1630’s, mr. rené descartes (1596 – 1650) would have gotten even more acclaim for his brilliant idea. of course, like many brilliant ideas, it is utterly simple in retrospect: take two number lines and put them at right angles to make the x-y coordinate plane. sometimes the x-y plane is called the “cartesian plane,” in honor of mr. descartes.

this was a truly remarkable breakthrough in mathematics because it created a deep connection between algebra (equations) and geometry (shapes & pictures): every equation with x & y gives us some kind of shape/picture in the x-y plane! these two branches of math had been completely separate and virtually unrelated until mr. descartes’ discovery. this paved the way for other, even more brilliant, interconnections between equations and shapes, such as mr. sir isaac newton‘s idea of calculus. fortunately, you don’t need to know about mr. newton’s ideas on the sat, but you do need to understand the big ideas of coordinate geometry.

useful gmat blogs for sat students

here are a few gmat blogs that discuss precisely the things you need to know for the sat:

1) quadrants of the x-y plane
chris also talks about this in his blog on coordinate geometry basics.

2) special properties of the line y = x
this discusses some of those quick tricks that the sat-writers love!

3) distance between two points
let your friend and mine, mr. pythagoras, convince you that memorizing a distance formula is a huge waste of time!

4) lines & slopes in the x-y plane
an important topic.

5) midpoints, parallel & perpendicular lines
another important topic.

if you want some practice questions, this next blog has some additional questions that could be on the sat:

6) coordinate geometry practice questions
ignore question #2: it is data sufficiency question, a question type unique to the gmat; you don’t need to know about these at all for the sat. just skip that question.

if you really want to challenge yourself, take a look at this one:

7) challenging coordinate geometry practice questions
if you can handle those eight questions, you will be able to handle anything the sat will throw in your direction about coordinate geometry.

summary

finally, here’s an sat practice question on coordinate geometry.

let us know about your experience of reading gmat blogs to prepare for the sat!

the post big ideas of sat coordinate geometry appeared first on magoosh blog | high school.

luckily, matrix questions are quite rare on the act (you might not see a single one throughout the 60 questions on the act math test!), and require only basic addition and subtraction skills (or *cough* a ti-83 calculator *cough*) to solve. if you can add 5 and 6, you’ll be fine!

act matrix basics

the word matrix refers to a rectangular-looking box filled with numbers arranged in rows and columns each number in the matrix is called an element. when it comes to matrices on the test, you really just need to know the basics:

first of all, matrices are referred to by their dimensions.

matrices can come in all sorts of sizes. we write the size of a matrix first with its number of rows and then its number of columns.

this, for example, is a 3 x 2 matrix. 3 rows and 2 columns:

scalar multiplication

you might see a question on the act that asks you do what is called scalar multiplication. this means that we are multiplying a matrix by an ordinary number. every entry inside the matrix just gets multiplied by that number.

matrix addition

you might also see a question on the act on adding or subtracting matrices. important note: you can only add or subtract two matrices of exactly the same dimensions. we add or subtract matrices by adding or subtracting the corresponding numbers (the numbers that are in the same “spot” on each matrix).

notice that the size of the matrices is the same, and that each element in the first matrix is added to the corresponding element in the second matrix to get the corresponding element in the third matrix. -3 + -2 = -5.

with that understanding, we can solve for x by writing a simple equation: x + 9 = 11; x = 2.

multiplying matrices

all right, i saved the trickiest for last. you can only multiply matrices if the number of columns in first matrix equals the number of rows in the second matrix. when we multiply matrices, the product matrix will have the same number of rows as first matrix and the same number of columns as the second. for example, the product of a 2 x 3 matrix and a 3 x 2 matrix would be a 3 x 3 matrix.

in order to find the entry in the first column, first row of the product matrix, we are going to multiply each number in the first row of the first matrix by its corresponding number in the first column of the second matrix and add the products together.

for the entry in the second row, second column, we will do the same thing with the second row of the first matrix and the second column of the second matrix:

in order to completely fill out this matrix, we would have to do 9 calculations, but don’t fret, the act is more likely to give you smaller matrices or matrices with more zeros or ask you to find just one entry in a matrix.

let’s look at a question involving subtracting matrices.
 

guided act matrix practice question

what is the sum of x and y?

to answer this, we will set up two equations to solve for x and y independently.

16 – 2 = x and 3 – (-4) = y

14 = x and 7 = y

their sum is 14 + 7 = 21.

sometimes a variable will be introduced to stand for an entire matrix. let’s look at an example:

carefully subtract each corresponding element.
 

the answer

now try a practice question on your own!
 

more act matrix practice questions

what is the solution to the matrices?

(a)

(b)

(c)

(d)

(e)
 

the answer & explanation

the correct answer is (a).

to subtract matrices, subtract each corresponding element, as shown above. remember that subtracting a negative number becomes addition.

with practice, you’ll see that the math involved in act matrix questions is rarely challenging – just keep the rows and columns lined up and you can’t help but get them right!

the post matrices on the act appeared first on magoosh blog | high school.

“i could remember more if i weren’t so stressed.”

do these sentiments sound familiar?  it seems our electronically driven culture recognizes two basic human energy states —- manic and stressed while on task, and then completely blitzed and unfocused afterward.  popular junk food serves to intensify this energetic dichotomy even further.  of course, neither of these states is optimal for mastery on the sat.

the best state both for learning & remembering and for performing under pressure is relaxed yet energized and focused.  while this state was common in other times and places, it’s hard for modern americans even to imagine it.   suppose i told you there was a way to get into that state, totally organic, with only beneficial health effects.  no, it’s not the newest thing — it’s closer to the oldest thing!  for over three thousand years, throughout the world, people have practiced meditation, and over time, this practice allows people to build focus and mental clarity while reducing stress.  trying meditation a couple times probably will not make a big difference, but if you could commit to this practice at this point in your life, the benefits over time would be considerable.

suppose your sat is coming up soon.  even if you start meditating every day, it probably won’t be long enough between now and your sat to have much effect.  well, some of the beneficial effects of meditating simply come from deep breathing.  when you breathe long, slow, deep breaths, this sends a message of relaxing and releasing stress to the body.  (by contrast, if your goal it to make yourself as stressed as possible, it will help to make your breathing rapid and shallow.)  you can practice this kind of silent deep breathing anywhere — in the car, waiting on a line, sitting in class, etc.   if the breath is particularly deep (i.e. both belly & chest expanding with air), then the increased oxygen in your blood will help to keep you awake and alert.  experiment with this, and notice how it shifts both your energy and your state of mind.

if you are even more ambitious about releasing stress, and building focus, i have some particularly challenging recommendations.

(1) eat well.  in particular, avoid high-sugar foods and anything with high-fructose corn syrup.  make a habit of drinking lots of plain ordinary water.

(2) get as much regular sleep as possible.  in an eight-hour stint of sleep, the greatest rem period is in the last hour, and that’s when the brain encodes memory.  if you skimp on sleep, you can use energy drinks to feel awake, but you can’t replace the opportunity to increase what your brain remembers and knows.

(3) avoid external excitement.  you may recall a diminutive zen master who said: “adventure, heh!  excitement, heh! a jedi craves not these things!”  you see, excitement and stress run on more or less the same brain circuitry, so the more you excitement you give yourself, the more stress you invite.   explore what it would mean to pursue appreciation rather than excitement.

(4) minimize any entertainment that involves watching an electronic screen.  believe it or not, the visual cues of such a screen stress your body.  when you do relax, pursue non-plugged-in forms of recreation: exercise and stretching, creativity and the arts, or time in nature.

yes, those recommendations would be hard to put into practice.  they are out of the ordinary.  here, i would remind you of the great law of mediocrity: if you do what everyone else does, you will probably wind up with the results that everyone else gets.   if you want standout, extraordinary results, say, on the sat, then you have to pursue extraordinary preparation such as these.

the most fascinating thing about all of this — meditation, deep breathing, and the other recommendations — these will help you not only on the sat, but in most other academic endeavors in your future.  and, you’ll be happier!

the post how to relax for the sat appeared first on magoosh blog | high school.