PSATSAT National Merit Scholarship Qualifying Test

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PSAT/SATNational Merit Scholarship Qualifying
Test
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How the Math PSAT is set up

• Unique type questions
• Funny symbol questions
• Ex if ab (ab)(ab) then 34 ?
• Quantitative Comparison
• column A column B questions
• Two 25-minute sections
• Regular Multiple Choice
• Special Multiple Choice with
• 12 Quantitative comparison
• 8 Grid in sections
• Typically there are
• 10 easy 19 medium 11 hard questions
• 12 Arithmetic 17 Algebra I 11 Geometry questions

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Scoring for the PSAT

• Scores range from 20 to 80
• A typical student that has not completed Geometry
  would like to score in the 40s
• A good score for a student that has completed
  Geometry would be 60 to 80
• National Merit scores are typically at least 200
  points from the 3 combined sections (this changes
  from year to year)
• For questions that are omitted you receive no
  points. Each correct answer earns one point. If
  you cannot make an educated guess then omit the
  question and move on. There is a penalty for
  guessing!
• Free-response/grid in questions there can be
  more than one answer possible no penalty for
  guessing in this section.
• Grid-in answer sheet
• Grid 2 ½ 2/3 204
• Note .66 or .67 are not as
• accurate as .666 or .667

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Sample questions

• Arithmetic questions
• 1. A total of 235 chairs were needed for the
  graduation ceremony at a local high school. A
  rental company was able to supply 128 chairs. If
  a local club that owned a total of 300, loaned
  the school the additional chairs, then how many
  chairs did the club have left?
• A. 65 B. 107 C. 128 D. 172 E. 193
• Answer E
• 2. Based on the chart above, which of the
  following is the closest to the wind chill
  temperature when the air temperature is 15o F and
  the wind speed is 20 miles per hour?
• A. -3 B. -13 C. -17 D. -20 E. -34
• Answer C Estimating would give either c or d
• 3. Which of the following numbers, when
  subtracted from 10, will give a result that is
  closest to 0?
• A. Ö 2 B. Ö 10 C. Ö 59 D. Ö 82 E. Ö 99
• Answer E

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Sample Questions--Arithmetic

• Five cars, each of a different color are to be
  parked in a row of 5 parking spaces. The number
  of possible arrangements of the 5 cars in the 5
  spaces is w. One of the cars is red.
• Column A Column B
• The number of possible
• arrangements of the 5 w
• cars in the 5 spaces if 5
• the red car is in the
• center space.
• Answer C
• If ½ of the number of white mice in a certain lab
  is 1/8 of the total number of mice and 1/3 of the
  gray mice is 1/9 of the total number of mice then
  what is the ratio of white mice to the number of
  gray mice?
• A. 1627 B. 23 C. 34 D. 45
  E. 89
• Answer C

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Algebra I questions

• From simple solutions to exponents, to factoring
  quadratics, algebraic fractions, exponents,
  linear equations, interpreting graphs, and word
  problems
• Review algebra I when in doubt pick numbers
  when picking numbers dont forget special rules
  for negatives, fractions, zero, one and one.
  If ratios or fractions are involved, pick the
  common denominator as your choice! If even or
  odds choose 3,4,and 5 keep the numbers small if
  possible!
• Dont jump straight to the calculator these
  questions are designed to not need a calculator
  and to cancel easilyif using a calculator, be
  careful of how you enter numbers in the
  calculator (-3)2 versus -32
• Funny symbol questions can be in each section
  from standard multiple choice to QC to Grid-in
• Example
• For all integers a, b, c let
• Column A Column B

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Algebra I samples

• Funny symbolsdont let them scare you just plug
  in the values
• 1. For all positive integers x and y , if is
  defined by xy (x y)-2 then (34)5
• 4 B.6 C. 8 D. 10 E. 58
• Answer B
• Exponent rules
• 2. If m and x are positive integers and 5(x1)
  5m which of the following equals 5x
• A. m (x1) B. m C. 5 (m-1) D. m-1 E.
  m/5
• Answer B
• Grid-in question
• 3. If ax 3 and a 1 then what is y ?
• cy 4 c 2
  x
• Answer 2/3 or .666 or .667
• Factoring
• 4. x(x-1)(x-2) (x-1)(x-2)(x-3) 0

5. Grid in If n is an integer such that 60 lt 5n
5 lt 90 what is one possible value for
n? Answer 11ltnlt17
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Geometry

• 1. In the figure above, lines l and m are
  parallel, In terms of x and y, which of the
  following is equal to z?
• y-x B. 90 x y
• 180 yx D. 180 y x E. y x 180
• Answer A
• 2. In the figure above, segments AC and BD
  intersect at E. Segments AD and BC are parallel
  and AE EB, BC AD CE ED. If AD 4 and the
  area of triangle AED is 7 then AB ?
• A. 4 B. 5 C. 6 D. 7 E. 8
• Answer D

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Geometry coordinate geo

• 4. In the figure above, point P lies on a line
  with a negative slope. Which of the labeled
  points could also lie on the line?
• A. B. C. D. E.
• Answer A
• AB BC CD
• Column A Column B
• X Y
• Answer B

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Reviewpage 1

• Number properties integers, even integers, odd
  integers
• Usually shown as variables
• If you pick numbers remember the special
  properties of 1, -1, 0
• If you are checking out the QA (col. A / col. B
  type questions check for positive/negative/zero/po
  wers of 2
• For integersmake sure that if you get an
  fraction/decimal you round!
• Be careful with negatives and subtraction! Be
  sure that you distribute a negative outside a set
  of parenthesis!
• Be careful with negative numbers and exponents!
  -5 versus (-5) and - x versus -x
• Rational, Irrational numbers
• Most important irrational numbers for PSAT are
• Ö2 Ö3 and p
• Radicals rules
• (Ö2) (Ö2) 2
• No radicals in the denominator of a fraction
• Conjugates for rationalizing
• Add/subract like radicals

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Page 1.5

• Comparing fractions easiest way is to get
  common denominator.
• Convert a repeating decimal to a fraction
  .444 4/9 .3434 34/99
• Percent is part
• 100 of
  100 whole
• Ratios part to part (ie. Boys to Girls) 23
• To convert to part to whole
• ( such as Boys to total kids)
• add the ratios for the whole 2
  (23)
• Averages
• Average sum of terms/number of terms
• To find the average of evenly spaced numbers
• Example to find the average of the even
  numbers from 21 to 86 (2286)/2
• Given an average to get back to the whole
  Average X number terms original whole

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Review (page 2)

• Probability and counting
• Basic m ways for an event to happen and n ways
  for a second event then the probability of the
  two events happening is mn
• Example probability of 1 die getting a 5 and
  getting a heads on a coin (1/6)(1/2)(1/12)
• How many outfits can be made from 4 shirts and 7
  pants (4)(7) 28 outfits
• Counting Factorial
• How many ways can the letters math be
  rearranged 4!
• How many ways can the 6 books be rearranged? 6!
• What if the center book is always the same book?
  5!
• Exponents
• xnxmxnm (xn)mxnm
• xn / xmxn-m x-n 1/xn
• an2 / an a n2-n a2

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Review (page 3)

• Multiply polynomials foil dont forget the oi
• (3x 4) (2x 5) 6x2 23x 20
• Factor
• GCF -4x2 40 x - 100
• Difference of squares 16x2 25 0
• Trinomials
• Perfect trinomials 16x2 24x 9
• Quadratic coefficient of 1 x2 15x 36
• Quadratic coefficient other than 1 3x2 -14x 8
• Grouping rs 5r st 5t
• Quadratics
• ax2 bx c y
• if it equals y then we have a parabola with a
  maximum or a minimum value
• Zeros are where the graph crosses the x axis
• To solve use factoring or the quadratic formula
• ax2 bx c 0
• To solve use factoring or the quadratic formula

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Review page 4

• Linear equations Ax By C or y
  mx b
• Slope of a line m (y-y) / (x-x)
• Parallel slope the slopes are
• Perpendicular the slopes are negative
  reciprocals
• Point-slope formula (y-y1) m (x-x1)
• Standard form
• no fractions/decimals
• A cant be negative
• A,B, and C cant all be 0
• A,B, and C must be reduced
• X-intercept is when y 0
• plug 0 in for y and solve to get x gives the
  point (x,0)
• Y-intercept is when x 0
• plug 0 in for x and solve to get y gives the
  point (0,y)
• A Horizontal line
• has slope 0 and has no variable x in the
  equation
• A vertical line
• has no slope and has no variable y in the
  equation

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Review (page 5)

• Geometry
• Lines and angles
• Linear pairs add to 180
• Vertical angles are
• Complementary angles add to 90
• Supplementary angles add to 180
• Parallel lines
• Alternate interior angles are
• Corresponding angles are
• Same-side interior angles are supplementary
• Triangles? Area ½ bh (h must be
  perpendicular)
• The exterior angle of a triangle the sum of the
  two remote interior angles
• The interior angles of a triangle add to 180
• The base angles of an isosceles triangle are
  congruent
• The legs of an isosceles triangle are
• Equilateral triangles have all sides and all
  angles 60
• Similar triangles have equal angles and
  proportional sides
• If two sides are known for a triangle then the
  third side lies between the sum of the two sides
  and the difference of the two sides a-b lt
  third side lt ab

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Review (page 6)

• Pythagorean?
• c2 a2 b2
• Pythagorean triples (3,4,5 5,12,13 7,
  24,25 8,15,17 )
• 30-60-90 triangles ratio of ? x x Ö3 2x
  
• 45-45-90 triangles ratio of? x x x Ö2
• Trigonometry
• Sin A opp/hypot
• Cos A adj/hypot
• Tan A opp/adj
• Right triangle hypot leg
• leg its piece
• piece 1 altitude
• altitude piece 2
• Polygons
• Sum of the interior angles (n-2) (180)
• Sum of exterior angles 360
• Regular polygon all angles are and all sides
  are
• One interior angle (n-2)180/ n

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Review (page 7)

• Quadrilaterals
• Parallelograms
• Opposite sides are
• Opposite sides are
• Opposite angles
• Consecutive angles are supplementary
• Diagonals bisect each other
• Rectangles
• All parallelogram rules
• All angles 90
• Diagonals are
• (diagonals DONOT bisect the angles!!)
• Rhombi
• All parallelogram rules
• All sides are
• Diagonals create right triangles with
  Pythagorean trig etc. Because
• Diagonals are perpendicular
• Diagonals bisect the angles
• Squares

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Review (page 8)

• Circles
• Area pr2
  Circumference 2 p r
• Area of a sector A sector (x/360) pr2
• Length of a sector (x/360) (2p r)
• Radius and tangent are perpendicular
• Length of a chord rs tu
• Length of secants r s tu
• Two chords equidistant from center are
• Draw a radius and use right triangle rules if
  possible
• Angles
• central angle arc
• On ½ (arc)
• In ½ ( arc arc)
• Out ½ (arc arc)
• Opposite angles of an inscribed quad are
  supplementary
• Two angles that inscribe the same arc are
• Area formulas

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Notes
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Student handout Algebra I questions

• Funny symbol questions can be in each section
  from standard multiple choice to QC to Grid-in
• Ex) For all integers a, b, c let
• Column A Column B
• 1. For all positive integers x and y , if is
  defined by xy (x y)-2 then (34)5
• 4 B.6 C. 8 D. 10 E. 58
• 2. If m and x are positive integers and 5(x1)
  5m which of the following equals 5x
• A. m (x1) B. m C. 5 (m-1) D. m-1 E.
  m/5
• 3. Grid-in If ax 3 and a 1 then what
  is y ?
• cy 4 c 2
  x

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Student HandoutSample Arithmetic Questions

• 1. A total of 235 chairs were needed for the
  graduation ceremony at a local high school. A
  rental company was able to supply 128 chairs. If
  a local club that owned a total of 300, loaned
  the school the additional chairs, then how many
  chairs did the club have left?
• A. 65 B. 107 C. 128 D. 172 E. 193
• 2. Based on the chart above, which of the
  following is the closest to the wind chill
  temperature when the air temperature is 15o F and
  the wind speed is 20 miles per hour?
• A. -3 B. -13 C. -17 D. -20 E. -34
• Which of the following numbers, when subtracted
  from 10, will give a result that is closest to 0?
• A. Ö 2 B. Ö 10 C. Ö 59 D. Ö 82 E. Ö 99
• 4. Five cars, each of a different color are to
  be parked in a row of 5 parking spaces. The
  number of possible arrangements of the 5 cars in
  the 5 spaces is w. One of the cars is red.
• Column A Column B