## Wednesday, March 24, 2010

### IMPROVING ACT MATH, PART 2

Now that the first 30 Math questions are perfect, and the next 15 are within sight of "perfect," it's time to buckle down on the last 15 questions and possibly fill in some of that Algebra II that may have appeared earlier on the test.

ALGREBRA II
- rules of exponents - when multiplying the same base, add the exponents; to raise an exponent to a higher power, multiply (and don't forget the coefficient if it's in a parenthesis with the variable); when dividing the same base, subtract the exponents; A NEGATIVE EXPONENT CHANGES THE POSITION OF THE BASE, NOT IT'S POSITIVE OR NEGATIVE VALUE (this is the most widely missed question relating to exponents).
- graphs - recognize linear, quadratic, and third degree graphs and translations of each. Know how to calculate or recognize slope, X intercepts (these are solutions -- any ordered pair satisfies the equation but only the roots solve it), Y intercepts, and points of intersection (where two graphs meet). Practice telling a story that fits a combination graph.
- functions - recognize odd and even functions from their exponents. Use the vertical line test to prove that an equation is a function. Use both the vertical line test and the horizontal line test to prove that the inverse is also a function (or that the original function is "one-to-one.") Practice substitution and compositions of functions.
- conic equations - circle (Did you realize that this is also the Pythagorean Theorem?), ellipse, hyperbola, and quadratic, too.
- logs - basic rules for expanding and simplifying.
- sequence and series - arithmetic and geometric. Arithmetic has a common difference (add the same thing every time to get the next term) and geometric has a common ratio (multiply by the same thing every time to get the next term).
- probability - successes divided by possibles. (Do you see a similarity with percent change?) Probability questions are appearing in the first 30 questions on a regular basis these days.
- variation - direct ( Y = kX) and inverse ( Y = k/X).
- systems of linear equations - substitution and elimination methods are used to solve for one variable at a time.
- matrices - setting up from a system of equations, adding, multiplying, and expanding through scalar multiplication. (Here's a perfect place to use the calculator to it's highest purpose!)

GEOMETRY
- good news - most of the geometry used is actually part of the elementary curriculum AND you won't have to do any proofs.
- special right triangles - Pythagorean Triplets, 30-60-90, and 45-45-90. Although you can always use the Pythagorean Theorem to find these relationships, the ACT and other college entrance exams are TIMED and knowing that 5-12-13 is a triplet can save precious minutes.
- SohCahToa- sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.
- parallel lines and transversals - know which angles are congruent and which are supplementary.
- properties of quadrilaterals - parallogram, rectangle, rhombus, square, trapezoid, and kite.
- circles - all radii are equal. (This is the most valuable property of a circle and will usually figure into a solution if a circle is used in a question.) Central angle equals the arc, inscribed angle equals 1/2 the arc, and outside angle equals (the big arc minus the little arc) times 1/2. Lines tangent to the circle are perpendicular to the radius.
- sum of interior angles of a polygon - (the number of sides minus 2) times 180 and each angle in the regular polygon is that equation divided by the number of sides. In a later blog, we'll look at another way to get the same answer without the need to remember an equation.
- volume - (if the sides are perpendicular to the base) the volume is "the area of the base times the height." (if the shape is a pyramid or cone) divide the answer by 3. The equation for finding the volume of a sphere will probably be given in the question.
- similar triangles - all angles are congruent and sides are proportional -- set up congruent ratios.
- relation between lines, areas, and volumes - if the sides are in the ratio 2 to 3 (2/3 or 2:3) then the ratio of the areas is (2/3) SQUARED, and the ratio of the volumes is (2/3)CUBED. Lines are first degree, areas are second degree, and volumes are third degree.

TRIGONOMETRY
- good news - probably half of the questions that the ACT calls Trig are from the Geometry curriculum in many schools. Go back and review everything you know about triangles, especially those pesky word problems involving shadows and ladders leaning against buildings.
- Pythagorean Identities - yet a fourth application of the Pythagorean Theorem. It is no wonder that we frequently say it is the "most important equation in all of math"and luckily it's taught around the fourth grade. Sine^2 + Cosine^2 = 1.
- The Unit Circle - know your reference angles and what's positive in each quadrant. ACT trig questions will usually refer to the axes and angles in radians, so know how to convert from degrees if needed.

You might be amazed at all of the Math that you've learned through the years. There are probably only a very few concepts that you haven't studied, so preparing for the Math section is really refreshing your brain and bringing the useful information to the top where it can be accessed quickly on demand.