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PSAT: Math TestObjectives:1. To understand the PSAT test format andscoring scheme2. To practice the two types of PSATquestions
Towers of HanoiIn this ancient game,there are threepegs with a numberof rings on the firstpeg arranged bysize.
Towers of HanoiThe point of thegame is to move allof the rings fromthe first peg to thelast peg, but youcan only place asmaller ring on topof a larger ring.
Towers of HanoiYou cannot place alarge ring on top ofa small ring. Oh,and you want to beable to do this withthe least number ofmoves possible.Click on the baby to play.
Towers of HanoiThe point here is thatthere are manyways to move thedisks from one pegto another, butthere is often afaster, moreefficient way.This is also the casewith many PSATquestions.
About the PSAT Originally meant “PreScholastic Aptitude Test” Designed to measure verbaland mathematical reasoning– Measures ability, not knowledge(yeah, right!)– Gives students a chance topractice SAT-type questionsand qualify for NMS
About the PSAT 2 hour, 10 minute test in 5 sections The math portion presents questionsgrounded in high school math but asksthem in unconventional ways. Test scores range from 20 – 80 on eachsection. Math section Averages:– A score of 48.9 is average for a Junior (2010)– A score of 44 is average for 10th (2010) To convert to a SAT score, add a zero
A Junior Year Test PSAT is considered a Junior year test– You may not recognize all of the mathematicson the test!– No Algebra II questions (not until the SAT)
What’s on the Test? Critical Reading: two 25 minute sections(48 MC questions) Mathematics: two 25 minute sections (38questions) Writing: one 30 minute section (39 MCquesitons) Questions are arranged in increasingdifficulty in each section.
Math Section DecompositionTwo math sections:1. Multiple-choice: 202. Multiple-choice: 8Grid-in: 10Tests includes number operations, algebraand functions, geometry andmeasurement, statistics and probability.
ScoringDescriptionCorrect answers (regardless ofdifficulty)Skipped questionsIncorrect multiple-choiceIncorrect grid-inScore 10-1/40
Two FactsWhat does the previous scoring guide andthe fact that the questions are listed inincreasing difficulty tell you about testtaking strategy? Don’t waste time on the hard questionsyou can’t answer. It doesn’t pay to guess unless you caneliminate at least one answer choice.
Guessing?What is the probability that you wouldchoose the correct answer to a 5-choicemultiple-choice question? 1/5So, if you randomly guessed on 5 questions,you would expect to get 1 right ( 1 point)and 4 wrong (-4/4 -1 point).Thus, you’d have a net gain of 0 points.
Guessing? Part 2What is the probability that you wouldchoose the correct answer to a 5-choicemultiple-choice question, if you eliminated1 choice?1/4So, if you eliminated one answer choice on 4questions, you would expect to get 1 right( 1 point) and 3 wrong (-3/4 point).Thus, you’d have a net gain of 1/4 point.
Math StrategiesSixteen of these 17 PSAT mathstrategies were adapted fromBarron's Math Workbook forthe New SAT by Lawrence S.Leff, which I highlyrecommend (click the link forproduct info fromAmazon.com). The 17thstrategy comes from JohnKatzman of PrincetonReview.
Math Strategies: 1Use the test booklet as scratch paperPoints H, J, K, and M lie on the same line, inthat order. If HK 8, JM 11, and JK 5,what is the ratio of HJ to HM? This is an obvious one: sketch a picture anwork out the problem in the booklet
Math Strategies: 2Find the quantity the question asks forIf (5 – 3)(x – 1) 4, then x2 Many PSAT questions will not simply askyou to solve for x. They go a step furtherand ask for x2 or 2x – 3.– Notice that if you solved for x, you’d get 3,which is one of the answer choices
Math Strategies: 3Plug in numbers to find a patternWhen a positive integer k is divided by 5, theremainder is 3. What is the remainderwhen 3k is divided by 5? Just find a couple of numbers that whenyou divide it by 5, you get a remainder of 3.Then multiply them by 3 and divide by 5.What’s the remainder each time?
Math Strategies: 4Choose a convenient starting value when none isgivenThe current value of a stock is 20% less than itsvalue when it was purchased. By whatpercent must the current value of the stockrise to have its original value? If a problem gives you no numbers to workwith, just choose a convenient one.– Percents: Start with 100– Fractions: Start with the common denominator
Math Strategies: 5Combine or multiply systems of equationsIf 8a – b2 24 and 8b b2 56, then a b Rather than give you a simple linear systemof equations, PSAT will give you strangesystems with exponents and fractions.Usually you can solve these with a cleversubstitution or by simply adding or subtractingyour equations.– Watch what happens when you add the aboveequations
Math Strategies: 6Account for all solutions of a higher-degreeequationIf x2 4x, then x The highest power in an equation tells youhow many solutions you should have.– So the above equation should have 2answers, not one.
Math Strategies: 7Make organized lists and look for patternsWhat is the units digit of 335? If you try to put this into a calculator, it willgive you a number that’s so big, it has toconvert it to scientific notation: None of thedigits on the screen is the unit digit.– Make a list of powers of 3, starting with 31, andwatch as a pattern emerges.
Math Strategies: 8Redraw figures to scaleIn rectangle JKLM above, if JK KL, what isKLthe ratio of JH to KM?HMJNote: Figure is not drawn to scale. Pictures on the PSAT are not usuallydrawn to scale, so redraw them accordingto the information in the problem.
Math Strategies: 9Subtract areas to find the area of an [un]shadedregionCBKAIn the figure shown, what is the areaof the [un]shaded region if ABCD is asquare and AKD is a semicirclewhose radius is 4?D PSAT will often have question where youhave to find the area of some oddly shapedregion. Just find the area of the larger figure,and then subtract the area of some other partyou can easily find.
Math Strategies: 9Subtract areas to find the area of an [un]shadedregionCBKAIn the figure shown, what is the areaof the [un]shaded region if ABCD is asquare and AKD is a semicirclewhose radius is 4?D Note that the actual tip should be to subtractto find the area of a shaded region. It’s justthe program I used to draw the figure abovecould not easily shade the correct region, so Ijust changed the problem.
Math Strategies: 10Test numerical answer choices in the questionIf 22x – 1 32, then x If you run into an equation you can’t easilysolve, just test out the answers they giveyou.– Notice that the answers are always inincreasing or decreasing order.
Math Strategies: 11Change variable answer choices into numbersIf m n is an odd number when m and n arepositive integers, which expression alwaysrepresents an even number? If you can’t deal with the variables as givenin a problem, choose numbers that makesense for the problem. Then test out eachanswer choice with your numbers.
Math Strategies: 12Write an algebraic equationTicket sales receipts for a music concert totaled 2,160.Three times as many tickets were sold for the Saturdaynight concert as was sold for the Sunday afternoonconcert. Two times as many tickets were sold for theFriday night concert as was sold for the Sundayafternoon concert. Tickets for all three concerts soldfor 2.00 each. Find the number of tickets sold for theSaturday night concert. You can get by lots of PSAT questions without havingto write your own algebraic equations. Sometimes,however, a problem is best solve by just defining avariable or two and writing an equation to solve.
Math Strategies: 13Look at a specific caseThe perimeter of a rectangle is 10 times asgreat as the width of the rectangle. Thelength of the rectangle is how many timesas great as the width of the rectangle? This is like choosing a convenient startingvalue. If a question is framed in a generalnature, try picking a specific case to workwith. (Use inductive reasoning!)
Math Strategies: 14Draw a diagramAmy goes shopping and spends one-third of hermoney on a new dress. She then goes toanother store and spends one-half of themoney she has left on shoes. If Amy has 56left after these two purchases, how muchmoney did she have when she startedshopping? Since this question involves differentfractions, draw yourself a simple diagram thatbreaks the money up into thirds.
Math Strategies: 15Work backwardsSara’s telephone service cost 21 per monthplus 0.25 for each local call, and longdistance calls are extra. Last month, Sara’sbill was 36.64, and it included 6.14 in longdistance charges. How many local calls didshe make? If a question is too convoluted to workforwards, just work backwards.– Start with the final bill about and start subtractingoff all her fees.
Math Strategies: 16Adopt a different point of viewA jar contains 110 marbles of which 50 arered and 60 are green. The probability ofpicking a red marble from the jar withoutlooking is, therefore, . How many redmarbles must be added to the jar so thatthe probability of picking a red marble willbe ?
Math Strategies: 17The Joe Bloggs TechniqueJoe Bloggs is the average test taker. Whenanswering the easy questions, if he comes byan answer easily, it is more than likely right.By the time he gets to the last question, whichis supposedly the hardest, he knows that if heis able to get the answer easily, that answer isprobably wrong. So he marks out the “easy”answer and chooses something else. Formore information on this crazy technique,read the “Joe Bloggs Eats Doughnuts”worksheet.
Math Strategies: 17The Joe Bloggs TechniqueIn the figure above, what is the greatestnumber of nonoverlapping regions intowhich the shaded region can be dividedwith exactly two straight lines?
Math Strategies: 17The Joe Bloggs TechniqueSince it would be really easy to just draw 2intersecting lines that divide this into 4regions, that answer must be wrongbecause it’s too easy, and this is the lastquestion on the test!
Assignment PSAT sample questions and mathstrategies
Test numerical answer choices in the question If 22x – 1 32, then x If you run into an equation you can’t easily solve, just test out the answers they give you. –Notice that the answe
