tag:blogger.com,1999:blog-86601227321770298552017-09-18T05:19:39.931-07:00SAT Math 4 LifeTips for the math section of the SAT Reasoning Test as well as the SAT Math II subject test.Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-8660122732177029855.post-49492044598542376712012-02-07T09:56:00.000-08:002012-06-06T17:52:30.359-07:00Parity - Even and Odd IntegersParity is a topic that is only briefly touched upon in school. It is one of those topics that students need extra study on in order to do better on the math portion of the SAT.<br /><br /><b>Parity</b> just means whether an integer is even or odd. It is a property of an integer. For example, we say that 0 has even parity, and 1 has odd parity.<br /><br />Even integers: ..., -4, -2, 0, 2, 4, ...<br />Odd integers: ..., -3, -1, 1, 3, ...<br /><br />You should be able to tell quickly if an integer is even or odd. Just look at the last digit. If the last digit is even (0, 2, 4, 6, 8), the integer is even. If the last digit is odd (1, 3, 5, 7, 9), the integer is odd. When looking at the examples below, don't actually perform the operations! Try to sense quickly whether the integers are even or odd, and if the result should be even or odd.<br /><br />You need to know how parity behaves when two integers are added, subtracted, multiplied, or divided. A special case of multiplication is taking powers. In this post, I will go over the rules for these. Problems can get harder when variables or other complications are involved--this may be covered in a future article.<br /><br /><a name='more'></a><br /><h3> Addition and Subtraction</h3><br />The same rules hold for both addition and subtraction, so I will just show the ones for addition.<br /><br />even + even = even<br />even + odd = odd<br />odd + odd = even<br /><br />Put another way:<br /><br />Sum of two integers with the same parity (both even or both odd) = even<br />Sum of two integers with different parity = odd<br /><br />Examples:<br /><br />24 + 88 = even<br />-2 + 34 = even<br />51 - 28 = odd<br /><br /><br /><h3> Multiplication</h3><br />even * even = even<br />even * odd = even<br />odd * odd = odd<br /><br />Put another way:<br /><br />even * (any integer, even or odd) = even<br />odd * odd = odd (is only way to get odd result)<br /><br />Examples:<br /><br />2 * 9 = even<br />-2 * 6 = even<br />5 * 7 = odd<br />-5 * 3 = odd<br /><br /><br /><h3> Division</h3><br />You need to be careful about division when it comes to integers and parity.<br /><br />First of all, when you divide two integers, the result is not always an integer. You can't tell if the result is an integer only by knowing the parity of the two integers you are dividing!<br /><br />0 / 0 = undefined<br />0 / (non-zero integer) = 0<br /><br />For the following, assume division gives an integer:<br /><br />even / odd = even<br />even / even = could be even or odd<br />odd / odd = odd<br /><br />Examples:<br /><br />45 / 1 = 45 odd<br />45 / 3 = 15 odd<br />45 / 7 = not an integer<br /><br />12 / 2 = 6 even<br />12 / 4 = 3 odd<br /><br /><br /><h3> Powers</h3><br />even ^ (any positive integer not 0) = even<br />odd ^ (any positive integer) = odd<br /><br />Focus on the base, not the exponent! The result has the same parity as the base.<br /><br />Examples:<br /><br />2^4 = even<br />3^4 = odd<br />(-48)^23 = even<br />(-513)^4 = odd<br /><br /><br /><h3> Related facts</h3><br /><ul><li>2 is the only even prime number; all other prime numbers are odd</li></ul><div><br /></div><div><br /></div><img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/YKYQiRW9WXg" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2012/02/parity-even-and-odd-integers.htmltag:blogger.com,1999:blog-8660122732177029855.post-9526390149154115852012-02-06T03:56:00.000-08:002012-02-10T05:51:31.519-08:00TI-89 Titanium calculatorI am meaning to write a long, detailed post about the TI-89 Titanium calculator; however, I want to introduce it quickly for now. I will fill in the details later. In the future, I will also have posts dedicated to using this calculator, including custom programs for it.<br /><br />For some strange reason, the TI-89 calculator is allowed on the SAT Math Level 2 subject test. Having this calculator, knowing how to use it well, and having it equipped with the right programs are definitely huge advantages when taking the test. I say this from having spent 2 years teaching prep classes specifically geared towards using this calculator.<br /><br />You can find the calculator in many stores like Target, Best Buy, OfficeMax, etc. If you prefer shopping online and getting a better price (and want to help support my blog), check out the following:<br /><br /><div style="text-align: center;"><iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="http://rcm.amazon.com/e/cm?lt1=_blank&bc1=000000&IS2=1&bg1=FFFFFF&fc1=000000&lc1=0000FF&t=sm4l-20&o=1&p=8&l=as4&m=amazon&f=ifr&ref=ss_til&asins=B0001EMLZ2" style="height: 240px; width: 120px;"></iframe></div><img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/vZrU4zcOf_M" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2012/02/ti-89-titanium-calculator.htmltag:blogger.com,1999:blog-8660122732177029855.post-58190969274808729562011-04-08T22:43:00.001-07:002012-02-06T04:04:37.803-08:00College ConfidentialHere's a great place to get information about the SAT, ACT, college admissions, and more:<br /><br /><a href="http://talk.collegeconfidential.com/sat-act-tests-test-preparation/">http://talk.collegeconfidential.com/sat-act-tests-test-preparation/</a><img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/SvgJv6lUu6w" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/04/college-confidential_08.htmltag:blogger.com,1999:blog-8660122732177029855.post-26792996507498477602011-03-09T00:20:00.001-08:002012-02-06T04:03:51.008-08:00Perfect square trivia12^2 = 144<br />21^2 = 441 <br /><br />13^2 = 169<br />31^2 = 961<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/ZFi9vNBqxU0" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/03/perfect-square-trivia_09.htmltag:blogger.com,1999:blog-8660122732177029855.post-37652189591030640772011-03-09T00:08:00.001-08:002012-02-06T04:02:31.932-08:00Geometry ChallengeTake any parallelogram. Construct squares on each side of the parallelogram. Prove that the centers of the constructed squares form a square.<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/5xpSvsI8Ycs" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/03/geometry-challenge_09.htmltag:blogger.com,1999:blog-8660122732177029855.post-41614332714652027572011-03-08T23:27:00.001-08:002012-02-06T04:03:36.641-08:000.99999... = 1Let x = 0.99999...<br /><br />10x = 9.99999...<br /><br />10x - x = 9.99999... - 9 = 9<br /><br />9x = 9<br /><br />x = 1<br /><br />Do you believe?<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/xT312NA1RbQ" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/03/099999-1_08.htmltag:blogger.com,1999:blog-8660122732177029855.post-87806826397108146812011-03-08T18:03:00.001-08:002012-02-06T04:05:22.203-08:00Log problemlog 237.5812087593 = 2.375812087593 (approximately)<br /><br />What is the pattern here? Can you write an equation for this that 2.375812087593 solves? The equation has exactly one other solution. What is the other solution? Can you prove that these are the only two solutions?<br /><br />(This is beyond the scope of the SAT Reasoning Test, but is reasonably related to the SAT Math 2 subject test.)<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/1Qdeg_cQ2Mk" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/03/log-problem_08.htmltag:blogger.com,1999:blog-8660122732177029855.post-27397271570645219512011-03-08T17:59:00.001-08:002012-02-06T04:04:29.163-08:00Digits ProblemEach letter represents a digit. Solve for the digits and numbers.<br /><br />TWO + THREE + SEVEN = TWELVE<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/jEcVGuKxp7E" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/03/digits-problem_08.htmltag:blogger.com,1999:blog-8660122732177029855.post-86205393184922202722011-02-12T19:44:00.001-08:002012-02-06T04:04:05.175-08:00Perfect SquaresThere's a certain amount of memorization that should be done for the math portion of the SAT. Relying on just the formulas given on the test is not enough (those are just a small, incomplete subset of what you need to know, and you shouldn't need to refer to them while taking the test as you should have them memorized). It's possible to get away with a lot using your calculator, but it's better not to.<br /><br />In school, you should have learned perfect squares up to at least 15^2 or 16^2. These are also the ones you should know for the SAT. The more you know, the better. If you don't know these perfect squares now, take the time to memorize the ones you don't know so well. (At the least, know how to generate a short list of them on your calculator very quickly--but memorization is the better choice.)<br /><br />Here's a list up to 20^2.<br /><br />1^2 = 1<br />2^2 = 4<br />3^2 = 9<br />4^2 = 16<br />5^2 = 25<br />6^2 = 36<br />7^2 = 49<br />8^2 = 64<br />9^2 = 81<br />10^2 = 100<br /><br />11^2 = 121<br />12^2 = 144<br />13^2 = 169<br />14^2 = 196<br />15^2 = 225<br />16^2 = 256<br />17^2 = 289<br />18^2 = 324<br />19^2 = 361<br />20^2 = 400<br /><br />There are many tricks and patterns with perfect squares, but I'll leave that for another time.<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/IpJJQvS6GOQ" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/02/perfect-squares_12.htmltag:blogger.com,1999:blog-8660122732177029855.post-59231128225789769102011-02-08T00:16:00.000-08:002012-02-06T04:04:16.892-08:00Isosceles Triangles - calculating area (given sides)This is an essential skill for the SAT. It is commonly needed to directly solve a math problem on the SAT or as an important step in a larger problem.<br /><br /><u><b>Definition:</b></u> Isosceles triangles are triangles that have at least 2 sides of equal length.<br /><br />There are many ways to find the area of various types of triangles. In this post, I will show you how to calculate the area of an isosceles triangle given the lengths of the 3 sides. The method illustrated is relatively straight-forward and is used by many students.<br /><br /><u><b>Given:</b></u> isosceles triangle with 3 given sides.<br /><br /><u><b>To find:</b></u> area of triangle (height is by-product).<br /><br /><u><b>Method:</b></u><br /><br />1. Draw an altitude from the vertex between the 2 equal sides to the opposite side.<br /><br />This opposite side serves as the base corresponding to the altitude just drawn. The base is now split in half. The isosceles triangle is split into 2 congruent right triangles (can you prove this?).<br /><br />2. Next, find the length of this altitude using the Pythagorean theorem.<br /><br />3. Plug & chug into A = (1/2) b * h.<br /><br /><u><b>Example:</b></u><br /><br />A triangle is given with sides 5, 5, 8. Choose the side of 8 to be the base. Draw the altitude to this base. Each of the right triangles formed has a hypotenuse of 5 and one leg of 4 (half of the side of 8). The other leg is the same as the altitude of the original triangle. It has length 3 by the Pythagorean theorem (3^2 + 4^2 = 5^2). Thus, the original triangle has base 8, height 3, and area (1/2) * 8 * 3 = 12.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/_PltDPkxQQ3A/TVEGQLoUGZI/AAAAAAAAAHA/e-gCvjtW-bI/s1600/isosceles+triangle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="132" src="http://2.bp.blogspot.com/_PltDPkxQQ3A/TVEGQLoUGZI/AAAAAAAAAHA/e-gCvjtW-bI/s400/isosceles+triangle.png" width="400" /></a></div><br /><br /><br /><u><b>Note:</b></u> Equilateral triangles are isosceles. You can find their areas in the same way. The 2 right triangles you construct will be 30-60-90 triangles. The height of the equilateral triangle will then be x * sqrt(3)/2, where x is the length of one of the sides. The area of the equilateral triangle is then x^2 * sqrt(3) / 4. These formulas for the height and area of an equilateral triangle are worth memorizing because equilateral triangles are so common on the SAT.<br /><br /><u><b>Practice:</b></u><br /><br />1. Find the height and area of a triangle with sides 13, 13, 10. (Area is 60.)<br /><br />2. Find the height and area of a triangle with sides 25, 25, 30. (Area is 300.)<br /><br />3. Find the height and area of a triangle with sides 3, 3, 3. (Area is 9 * sqrt(3) / 4.)<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/IlI76yeJV4E" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/02/isosceles-triangles-calculating-area.htmltag:blogger.com,1999:blog-8660122732177029855.post-30497435911818457072011-02-07T23:54:00.000-08:002012-01-29T02:09:32.591-08:00First postHi,<br /><br />I will use this blog for a variety of things. Probably, mostly for SAT math tips, test-taking strategies, and study strategies for now...<br /><br />WARNING on math: I could get very technical about the way I write math, but I try not to in order to make things easier to understand. For instance, I will write things like "triangle with sides 3, 4, 5" when I really mean the sides have lengths of 3, 4, and 5. This is also the way I tend to speak in person.<br /><br />I'm not that careful about grammar, but feel free to correct me.<br /><br />I also tend to edit things a lot, so you might see things change after you read it the first time.<br /><br />Please feel free to ask questions or add other comments.<img src="http://feeds.feedburner.com/~r/blogspot/WHRqa/~4/E3_pMQoG3vw" height="1" width="1" alt=""/>Seanhttp://www.blogger.com/profile/05749984493242487817noreply@blogger.com0http://satmath4life.blogspot.com/2011/02/first-post.html